Abstract

Self-coherent detection with interferometric field reconstruction aims at retrieving the complex-valued optical field (amplitude and phase) by digitally processing delay interferometer (DI) measurements, in order to realize a differential direct detection receiver with capabilities akin to that of a fully coherent receiver with polarization multiplexing, albeit without requiring a local oscillator laser in the receiver. Here we introduce a novel digital recursive algorithm capable of accurately reconstructing the optical complex field (both amplitude and phase) solely from the quadrature DI outputs, eliminating the AM photo-detector branch. We analyze a key impairment namely the accumulation of errors and fluctuations in the reconstructed amplitude and phase due to ADC quantization noise, recirculating in the recursion. We introduce signal processing measures to effectively mitigate this noise impairment leading to a potentially practical self-coherent receiver, demonstrated in this paper for a single polarization. We also investigate the range of applicability of self-coherent detection concluding that it is most suitable to relatively low baud-rate systems such as passive optical networks, for which application the self-coherent receiver outperforms the coherent homodyne receiver due to its improved laser noise tolerance, obtained due to the removal of the optical local oscillator.

© 2012 OSA

1. Introduction

In order to meet the ever increasing demand for telecommunication capacity, fiber-optic transmission has been evolving dramatically over the past decade. Several years ago, just prior to the renaissance in coherent detection, an evolutionary transition took place from direct detection to differential direct detection (DDD), such as Differential Binary/Quaternary Shift Keying (DBPSK/DQPSK). This enabled extension of the long-haul transmission rates from 10 Gb/s to 40 Gb/s, at the expense of requiring a more complex receiver (Rx) optical front-end based on delay interferometers (DI). The brief DDD epoch was followed by the recent disruptive introduction of optically coherent detection of the complex field coupled with digital signal processing (DSP), enabling extension of the bitrate to 100 Gb/s and beyond.

Self-coherent (SC) detection with interferometric field reconstruction (FR) [16] aims at retrieving the complex-valued optical field (amplitude and phase) by digitally processing the DI outputs, in order to provide capability akin to that of a fully coherent receiver, albeit without requiring an optical local oscillator (OLO) laser in the receiver. Eliminating the OLO while essentially retaining the advantages of coherent detection would enable all the performance advantages of coherent detection at low cost.

For a recent review of SC techniques see for example Chapter 1 in [7], titled “Coherent, SC and Differential Detection Systems”. Some of the prior SC schemes were further equipped with a power measuring photo-diode, enabling to directly reconstruct the amplitude of the optical field from an optical power measurement, while the phase was retrieved by digitally integrating the phase difference samples, as measured at the DI outputs. Another prior work [3] attempted to eliminate the amplitude photo-detection by introducing rudimentary processing of pairs of successive DI output amplitudes. Unfortunately, such approximate algorithm does not provide a reliable field amplitude estimate. Yet another approach to SC field reconstruction extracts analog FM demodulation based on a single DI with very short delay [2]. These schemes might not be suitable to QAM detection. Here we introduce a novel digital recursive algorithm accurately reconstructing the optical complex field (both amplitude and phase) solely from the IQ differential measurements, eliminating the Intensity Modulated Direct Detection (IM-DD) photo-detector branch, conceiving, to the best of our knowledge, the first SC Rx capable of supporting 16-QAM transmission. We model for the first time a key impairment which severely impacted prior SC FR schemes, namely the accumulation of fluctuations in the reconstructed amplitude and phase due to ADC quantization noise, recirculating in the recursive FR algorithm. We theoretically analyze and numerically simulate this noisy random walk of the reconstructed field, and introduce additional signal processing measures to effectively mitigate this critical noise impairment leading to a potentially practical SC receiver, demonstrated in this paper for a single polarization. The novel SC receiver combines fractional-delay IQ DI, twice oversampling and a carrier recovery (CR) system of the Multi Symbol Delay Detection (MSDD) type [8,9], modified to include an adaptive Normalized Least Mean Squares (NLMS) based Automatic Gain Control (AGC) function, which is essential in mitigating the amplitude random walk inherent in the SC FR operation. Moreover, we propose and simulate a counter-measure to the issue of division by zero, which might otherwise severely impair the SC receiver by causing occasional outages.

Our impairments analysis indicates that SC detection is best applicable to relatively low-baud-rate low chromatic dispersion links, such as in next generation Passive Optical Networks (PON) aiming for 1 Gb/s sustained data rate per user. For such optical access applications, laserless SC optical network units would be highly cost-effective. Remarkably, we show that, in this low baud-rate operational regime, the SC receiver significantly outperforms a comparable fully-coherent receiver and we explain the origin of this advantage.

The paper is structured as follows. Section 2 reviews IQ DI structures. In section 3 we introduce our novel complex-valued recursive SC FR algorithm. Section 4 performs a numerical accuracy analysis of the FR algorithm, elucidating the mechanisms of cumulative noise runoff (random walk of amplitude and phase perturbations). Section 5 introduces the structure of the proposed SC twice-oversampled single-polarization receiver with MSDD CR. Section 6 simulates the quantization noise random walk at the FR output. Section 7 simulates single polarization SC 16-QAM transmission. Section 8 addresses the division by zero or by extremely low values exception. The concluding Section 9 provides perspective for the SC detection. The abbreviations used in this paper are listed in an Appendix, for the readers’ convenience.

2. IQ DI realization and modeling

The SC Rx pursued in this paper is essentially a digitally assisted direct-detection receiver equipped with an IQ interferometric front-end, e.g. comprising a pair of DIs for each polarization and possibly an extra IM-DD branch. The Rx front-end outputs are analog-to-digital converted, then digitally processed in a field reconstruction (FR) module, extracting the complex field samples from the front-end digitized outputs, by suitable algorithms. In this respect, the “self-“ designator in SC detection indicates coherent-like operation without an optical local oscillator. The resulting complex field estimate may be further processed just as in a conventional coherent receiver, in order to mitigate optical channel impairments.

We consider two alternative SC Rx front-end structures. The first one (Fig. 1(a) ) comprises a pair of DIs (per polarization) referred to as I and Q DIs, differing by 90 deg in their phase biases (Fig. 1a). This is the same structure as used in a DQPSK DDD Rx front-end. The second SC front-end shown in Fig. 1(b), proposed in [10], might be more convenient to use in practice as it is based on a ubiquitous coherent technology component namely the 90 deg optical hybrid. In the sequel we shall refer to the first configuration, however the two SC front-ends are equivalent, hence all conclusions equally apply to the one in Fig. 1(b).

 

Fig. 1 SC receiver front-end alternatives. (a): An IQ DI realization consisting of a pair of delay interferometers in quadrature. (b): An equivalent 90 deg hybrid-based realization of the IQ DI.

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The balanced photo-detector photo-current outputs of the two DIs are expressed, up to a constant, as follows in terms of the received field complex envelope ρ˜(t) at the DI inputs

I(t)=Re{ρ˜(t)ρ˜*(tτDI)}=ρ(t)ρ(tτDI)cos[ρ˜kρ˜(tτDI)]Q(t)=Im{ρ˜(t)ρ˜*(tτDI)}=ρ(t)ρ(tτDI)sin[ρ˜kρ˜(tτDI)]
In terms of notation, an undertilde indicates a complex-valued quantity – removing the undertilde indicates the magnitude or modulus of that quantity, e.g. ρ(t)|ρ˜(t)|. An alternative SC Rx front-end structure (Fig. 1(b)) consists of a 90 deg coherent hybrid, just like used in coherent detection but with the LO removed and replaced by a τDI-delayed replica of the received signal. It is readily shown that the I(t) and Q(t) currents at the output of the balanced photo-diode pairs of the hybrid are also given by the same Eqs. (1), thus the two IQ DI realizations of Fig. 1(a,b) are equivalent.

Let us now define a complex-valued IQ DI analog output, q˜(t), as follows:

q˜(t)=I(t)+jQ(t)=ρ˜(t)ρ˜*(tτDI)
where we used (1) to derive the last expression. Now, digitally sampling the IQ DI complex-valued output at times t=kτDI yields the samples,
q˜kq˜(kτDI)=ρ˜(kτDI)ρ˜*((k1)τDI)
Denoting the input optical field samples by ρ˜k=ρ˜(kτDI), the following compact relation then describes the discrete-time transformation performed by the IQ DI:
q˜k=ρ˜kρ˜k1*
Alternatively, introducing the samples IkI(kτDI),QkI(kτDI)of the I and Q individual IQ DI outputs, we have |
Ik=Reρ˜kρ˜k1*=ρkρk1cos(ρ˜kρ˜k1)Qk=Imρ˜kρ˜k1*=ρkρk1sin(ρ˜kρ˜k1).
q˜k=Ik+jQk=Re{ρ˜kρ˜k1*}+jIm{ρ˜kρ˜k1*}=ρ˜kρ˜k1*=ρkρk1ej(ρ˜kρ˜k1)
We note that for proper operation the two DIs should be in perfect quadrature and have the same gains in their electro-optic analog detection chains or equivalently the 90 deg hybrid of Fig. 1(b) should be perfectly balanced. The degradation of SC detection due to “IQ imbalance” and its mitigation are outside the scope of the current paper. It is also important to have the ADC operate with no memory of prior samples. In the remainder of the paper we assume that our measuredq˜ksamples are perfect, described by Eq. (6) apart from post-detection noise at the IQ DI outputs, namely receiver noise and quantization noise in the ADC, thus we may write q˜k=q˜ko+n˜k where the superscript odenotes noiseless samples which would have been received in the absence of noise, and n˜k=n˜kRe+jn˜kImis a stationary complex-valued noise process with real and imaginary parts given by the post-detection (thermal and ADC quantization) noise processes affecting the respective I and Q samples. Notice that the noise process is not Gaussian as quantization noise is uniformly rather than Gaussian distributed.

3. Complex-valued recursive algorithm for SC field reconstruction

It is our objective to introduce, analyze and evaluate by simulation, a novel recursive optical field reconstruction technique for SC detection, providing precise field phase as well as amplitude retrieval without requiring a separate IM-DD branch. We propose to equip the SC Rx with a novel FR algorithm, in principle reconstructing both the field phase and magnitude without error, ideally assuming floating point processing and zero post-detection noise with an infinite number of bits in the ADC. As long as the field samples do not strictly cross zero, and quantization noise is negligible, this theoretical algorithm functions perfectly (whereas the field amplitude reconstruction algorithm [3] provides a gross estimate of magnitude even under ideal conditions).

2.1 Field Reconstruction problem statement

The input to our FR procedure is the ideal IQ DI complex output q˜kof Eq. (4), which is a complex representation of the two ideal DI outputs (for a particular polarization component). Evidently the sequence q˜k=ρ˜kρ˜k1* is a non-linear function of the field samples sequence, ρ˜k. Measuring q˜k=Ik+jQk, as formed from the samples Ik,Qkof the IQ DI outputs, we wish to reconstruct the complex samples ρ˜k=|ρ˜k|ejρ˜k of the received optical field at the input to the splitter feeding the IQ DI (Fig. 1), in effect, inverting the non-linear mapping ρ˜kq˜k. The inverse mapping q˜kρ˜kis provided by the FR algorithm.

2.2 Brief review of previous field reconstruction methods using delay interferometers (DI)

Previous digital FR approaches most similar to ours were pioneered by N. Kikuchi [1,4] and X. Liu [3]. Advances until 2010 are summarized in a review article in [7]. Heretofore the field reconstruction problem has been approached in polar form, separately addressing the magnitude and phase reconstruction problems. In our notation, the DI output (6) is converted from I-Q cartezian form to a polar (r,ϕ)representation, extracting magnitude and phase of the complex DI output,

qk=Ik2+Qk2=ρkρk1,q˜k=arctanQkIk=ρ˜kρ˜k1
and processing magnitude and phase samples separately. Treating the phase first, as q˜kis essentially a discrete-time derivative of the phase ρ˜k of the field (difference of successive samples), then the phase of the field may be, in principle, simply reconstructed by digital integration (accumulation), passing the IQ DI output angle through the cumulative recursion,
ρ˜^k=q˜k+ρ˜^k1=k=0kq˜k
with some arbitrary initial condition, ρ˜^0, e.g. ρ˜^0=0. The field phase is seen to be reconstructed up to an unknown additive constant, corresponding to the difference ρ˜0ρ˜^0between the actual and assumed initial fields. For this unknown additive phase-shift to not degrade receiver performance, it is essential to use a differential precoder (DP) in the transmitter (Tx), matched by a corresponding carrier recovery method in the receiver [8], [9]. Here we investigate QAM transmission, adopting the QAM-oriented DP method introduced by Kikuchi (passing on the QAM magnitude while differentially encoding the phase), which DP method is referred to as magnitude-preserving DP in [8] [9]. The carrier recovery method we used in our self-coherent receiver is MSDD, which is insensitive to an arbitrary phase offset in the received field, as also used in [4]. See also [11] for a similar carrier recovery method.

As for evaluation of the field magnitude, ρk, in the approach of N. Kikuchi no attempt is made to estimate it from the DI output measurements but the hardware is made more complex in order to enable separately detecting the field magnitude: light is split to an additional intensity modulation (IM) measurement branch where a photo-receiver followed by an ADC measures the samples of the optical power, Pk=|ρ˜k|2=ρk2, simply obtaining ρkby taking the square root of the optical power: ρ^k=Pk. Thus, the overall field estimate may be compactly expressed in terms of the three front-end measurements Pk,Ik,Qk, as follows:

ρ˜^k=Pkexp{jk=0karctan(Qk/Ik)}
It would be advantageous to simplify the SC Rx by eliminating the additional IM measurement branch. X. Liu [3] observed that the field magnitude information ρk is actually embedded in the magnitude qk=Ik2+Qk2=ρkρk1 of the complex DI output, and proposed the following simple algorithm to estimate the field magnitude directly from the DI outputs Ik,Qk without necessitating a third IM measurement branch, as follows:
ρ^kρkρk1qk=(Ik2+Qk2)1/4
Evidently this algorithm merely provides a gross approximation of the actual magnitude of the field, breaking down when the field varies rapidly, yielding reasonable results just for slowly varying fields. An effective amplitude measurement method from the outputs of the IQ DI structure of Fig. 1 does not currently exist. Here we set out to precisely estimate the full complex ρ^˜k just in terms of Ik,Qk, without resorting to a third IM branch, thus eliminating the Pkmeasurement. Our proposed algorithm accuracy is just limited by the post-detection noise and numerical accuracy effects, i.e. the number of bits used in the ADC and complex arithmetic operations. Moreover, in our approach we do not have to perform cartesian (IQ) to polar transformations. We avoid separate processing of magnitude and phase. Rather, our FR algorithm directly operates on the received complex-valued samples, applying complex-valued signal processing operations to directly generate the complex field samples in the IQ domain, as described next.

2.3 Novel field reconstruction algorithm based on recursive complex division

Our novel FR algorithm (Fig. 2 ) is strikingly simple, requiring a single complex division, yet somewhat tricky to comprehend, especially regarding the impact of initial conditions. The field samples are reconstructed by the following simple recursion, realizable just with a single recursive conjugate divider (RCD) performing division of its first complex-valued input by the complex conjugate of its second input:

ρ˜^k=q˜k/ρ˜^k1*witharbitraryinitialconditionρ˜^0
where ρ˜^kdenotes the estimate we generate for the true ρ˜k .

 

Fig. 2 The field reconstruction algorithm embedded in a schematic of a self-coherent receiver.

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The recursion (30) is simply derived by solving for the unknown ρ˜k in Eq. (4), q˜kρ˜kρ˜k1*, while assuming that ρ˜k1*is already known, from the previous recursion step. The impact of the initial conditions will be elaborated below. At first sight it seems that this algorithm must be strictly initialized with the proper initial condition ρ˜^0=ρ^0ejρ˜^0=ρ˜0 . Evidently, the Rx is not cognizant of the initial condition, hence it rather initializes the recursion Eq. (11) with an arbitrary non-zero value, ρ˜^0, say ρ˜^0=1. The lack of knowledge of the initial condition implies that the field will be reconstructed up to a multiplicative complex-valued constant, i.e. the precise amplitude scale will not be known, whereas the phase will be known up to an unknown additive constant. Actually, the situation is a bit more complicated, the statements just made separately apply to the even and odd polyphase subsequences of the reconstructed field, as shown next. For now, let us assume we have the correct values for both the magnitude and phase of the initial condition at k = 0 (a “genie” tells us the complex initial conditionρ˜0), thus, we precisely set the initial condition ρ˜^0=ρ˜0. Once properly initialized, it is straightforward to show that the recursion (12) precisely reconstructs the field forever. The FR algorithm Eq. (11) recursive steps are:

ρ˜^1=q˜1ρ˜^0*;ρ˜^2=q˜2ρ˜^1*;ρ˜^3=q˜3ρ˜^2*.....ρ˜^k=q˜kρ˜^k1*....
Step-by-step, starting with the “genie initial condition”, ρ˜^0=ρ˜0, we have ρ˜^1=q˜1ρ˜^0*=ρ˜1ρ˜0*ρ˜0*=ρ˜1 . Next,ρ˜^2=q˜2ρ˜^1*=ρ˜2ρ˜1*ρ˜1*=ρ˜2, ... etc.. So this algorithm, initialized by a genie works perfectly. Evidently, in practice the initial field sample ρ˜0 is known neither in magnitude nor in phase. Nevertheless, we show that even with arbitrary incorrect initialization, ρ˜^0ρ˜0, we nevertheless obtain a well-functioning end-to-end system, in conjunction with a carrier-recovery system and possibly other means to provide automatic gain control functionality.

Let us represent the initialization mismatch, i.e. the discrepancy between the initial condition arbitrarily assumed, and the actual initial condition, by the ratio g˜0ρ˜^0/ρ˜01 between the assumed and true initial condition. Let us then assess the effect of the incorrect initial condition, ρ˜^0=g˜0ρ˜0, differing from the actual ρ˜0 by the complex gain factor g˜01. Using q˜kρ˜kρ˜k1*, yields step-by-step:

k=1,2,3,4,...:k=1:ρ˜^1=q˜1ρ˜^0*=ρ˜1ρ˜0*g˜0*ρ˜0*=ρ˜1/g˜0*
i.e. we reconstructed the field at k = 1 up to a complex factor 1/g˜0*. Next,
k=2:ρ˜^2=q˜2ρ˜^1*=ρ˜2ρ˜1*(ρ˜1/g˜0*)*=ρ˜2ρ˜1*ρ˜1*/g˜0=ρ˜2g˜0
i.e. we now reconstructed the field at k = 2 up to an (inverse conjugate) complex factor g˜0.
k=3:ρ˜^3=q˜3ρ˜^2*=ρ˜3ρ˜2*(ρ˜2g˜0)*=ρ˜3ρ˜2*ρ˜2*g˜0*=ρ˜3/g˜0*
Thus, for k = 3 we are back to reconstruction up to 1/g˜0* as for k = 1. After one more step,
k=4:ρ˜^3=q˜3ρ˜^2*=ρ˜3ρ˜2*(ρ˜2g˜0)*=ρ˜3ρ˜2*ρ˜2*g˜0*=ρ˜3/g˜0*ρ˜^4=q˜4ρ˜^3*=ρ˜4ρ˜3*(ρ˜3/g˜0*)*=ρ˜4ρ˜3*ρ˜3*/g˜0=ρ˜4g˜0
i.e., for k = 4 we are back to reconstruction up to the g˜0 factor as for k = 2. The emerging pattern is that odd samples are reconstructed up to 1/g˜0* whereas even samples are reconstructed up to g˜0 (this may be readily formally proven by induction, for general k). Evidently, if g˜0=1, i.e., we just happened to start with the correct initial condition, then we would have perfect reconstruction. However, when starting with an arbitrary initial condition, g˜01, the even and odd polyphase subsequences experience two distinct complex gains:
{ρ˜^0,ρ˜^2,ρ˜^4,...,ρ˜^2k,...}=g˜0{ρ˜0,ρ˜2,ρ˜4,...,ρ˜2k,...}{ρ˜^1,ρ˜^3,ρ˜^5,...,ρ˜^2k+1,...}=g˜1{ρ˜^1,ρ˜^3,ρ˜^5,...,ρ˜^2k+1,...}whereg˜11/g˜0*
Interestingly, g˜1={1/g˜0*}=g˜0γ0, i.e. the reconstructed samples in both polyphases are identically phase-shifted with respect to the true phases, by the constant phase-bias
γ0g˜0:ρ˜^k=ρ˜k+γ0,k=0,1,2,3,4,....
Thus, the proposed FR recursion Eq. (12) reconstructs the field samples up to a fixed phase-tilt γ0, whereas the amplitudes of successive reconstructed field samples oscillate up/down.

As our SC system is based on differential precoding in the Tx as well as a generalized form of differential decoding in the Rx (MSDD carrier recovery [9]) the unknown but fixed phase-shift g˜0=γ0 added up to all reconstructed samples (stemming from phase error γ0 of the initial condition, ρ˜^0=ρ0g0exp{j(ρ˜0+γ0)}) is inconsequential, as it is cancelled out in the MSDD carrier recovery process.

The up/down oscillation of the reconstructed magnitudes of the successive even/odd samples is henceforth referred to as alternation effect – traced to the discrepancy between the magnitude of the initially set condition and the true magnitude. This effect amounts to having the even and odd polyphase subsequences of the reconstructed field samples ρ^kexperience fixed but different gain factors. Upon partitioning the field samples sequence into even and odd sub-sequences, each subsequence would experience scaling by a fixed gain factor, though the two fixed gain factors for the even and odd subsequences are different (in fact are inverses of each other). The alternation effect is not mitigated within the FR subsystem, but the alternating even/odd gain factors may be recalibrated in the subsequent receiver stages, by partitioning the incoming sequence of samples into even and odd polyphases, and separately processing the two polyphase sub-sequences. Each polyphase processing sub-module should have an automatic gain control (ADC) capability, properly rescaling the constellation prior to slicing. In section 5, we shall introduce an oversampling variant of the receiver which decimates the output of the FR, resorting to processing a single polyphase, thus simply eliminating the FR alternation effect. By using twice oversampling in the SC Rx, the sub-sequent 2:1 down-sampling extracts either the odd or even polyphase, and the alternation effect is mitigated.

4. Numerical accuracy analysis of the field reconstruction algorithm

The FR module essentially comprises a recursive divider. Representing the recursive divider of Eq. (12) in polar rather than cartesian (I-Q) form, and taking the magnitude and phase of the FR recursion Eq. (12), yields two separate recursions:

ρ^k=qk/ρ^k1;ρ˜^k=q˜k+ρ˜^k1
Thus, we have an accumulator for the phase and a recursive divider for the magnitude.

The post-detection noise accompanying the DI outputs was modeled at the end of section 2 as additive complex circular white noise superposed onto the two I and Q DI outputs. In this paper we further assume that the SC RX is optically pre-amplified and sufficient optical gain is provided such that the receiver is ASE beat-noise limited, i.e. the thermal noise is effectively negligible. Therefore, the main source of additive white noise in the I and Q DI outputs is ADC quantization noise, which amounts to a complex noise process n˜k being added to the noiseless complex DI output: q˜k=q˜ko+n˜k. We derive the cumulative noise runoff properties of the recursive divider, based on the equivalent concept of relative (normalized) noise.

Assume the input q˜kinto the FR module carries post-detection and processing noise including quantization distortion. Ideal noise-free quantities are denoted by a superscript o, and noise perturbations are defined as deviations between the noisy and noiseless quantities. The noisy input and output of the FR module are then expressed as,

q˜k=q˜ko+n˜kq=q˜ko(1+n˜kq/q˜ko)=q˜ko(1+η˜kq)ρ˜^k=ρ˜^ko+n˜kρ=ρ˜^ko+n˜kρ=ρ˜^ko(1+n˜kρ/ρ˜^ko)=ρ˜^ko(1+η˜kρ)
where normalized or relative noises were introduced as follows:
ηkqn˜kq/q˜ko=(q˜kq˜ko)/q˜ko;ηkρn˜kρ/ρ˜ko=(ρ˜kρ˜ko)/ρ˜ko
The magnitude of the reconstructed field is
ρ^k|ρ˜^k|=ρ^ko|1+η˜kρ|=ρ^ko|1+ηkρRe+jηkρIm|ρ^ko(1+ηkρRe)
The FR recursive division is expressed as follows:
ρ˜^kρ˜kρ˜k1*=q˜kρ˜^k1*=q˜ko(1+η˜kq)ρ˜^k1o*(1+η˜k1ρ*)=ρ˜^ko(1+η˜kqη˜k1ρ*)
where the first-order Taylor expansion 1+x1+y=1+xyxy+y2+... was used, retaining just the linear terms, as the second-order noise×noise terms are negligible.

Comparing Eq. (24) with the last expression in Eq. (21), we identify:

η˜kρ=η˜kqη˜k1ρ*
Recalling that η˜kqis the input noise into the FR module, whereas η˜kρis the noise in the reconstructed field at the FR module output, the recursion of Eq. (25) for the FR output noise corresponds to a discrete-time non-linear filter (non-linear due to the presence of the complex conjugation). However, by separating Eq. (25) into real and imaginary parts, the formulation reduces to two linear time-invariant (LTI) filters:
ηkρRe=ηkqReηk1ρRe;ηkρIm=ηkqIm+ηk1ρIm
The transfer function describing the recursion for the real part of the noise is obtained by taking Z-transforms, as follows,
Z{ηkρRe}=Z{ηkqRe}z1Z{ηkρRe}Z{ηkρRe}Z{ηkqRe}=11+z1=zz+1
This transfer function for the I-component of the relative noise has a zero at the origin and a pole at z=1. Similarly, the transfer function for the Q component of the noise is obtained as follows,
Z{ηkρIm}=Z{ηkqIm}+z1Z{ηkρIm}Z{ηkρIm}Z{ηkqIm}=11z1=zz1
seen to have a zero at the origin and a pole at z=+1. The different pole positions imply distinct noise filtering properties for the real and imaginary parts of the noise propagating through the FR module. The amplitude and phase perturbations are both independent-increments cumulative processes. For the phase, the variance linearly grows without bound. This phase degradation mechanism is similar to laser phase noise, which is also a cumulative Wiener random walk process. Decomposing the input noise terms into frequency components, it is apparent that the various frequency components experience different rates of noise accumulation. For the phase process, low frequency input phase noise components get strongly amplified. In fact a DC input phase perturbation grows without bound. However, for the relative magnitude, it is the frequency components in the vicinity of half the sampling frequency which get strongly amplified. In particular, a relative magnitude noise term at precisely half the sampling frequency tends to grow without bound. As long as the output SNR remains high, i.e. when the relative noise η˜kρ˜is small relative to unity, then the total output may be expressed in polar form in terms of the I and Q components of η˜kρ˜, as follows,
ρ˜^k=ρ˜^ko(1+ηkρ)=ρ˜^ko(1+ηkqRe+jηkqIm)ρ˜^ko(1+ηkqRe)ejηkqIm
i.e. the fluctuation η˜kqImin I-DI output contributes to the reconstructed field output amplitude noise, whereas the fluctuation η˜kqRe in Q-DI output contributes to the reconstructed field output phase noise. The recursion of Eq. (26) for ηkρIm amounts a discrete-time accumulator with impulse response, uk given by the discrete-time step function. The output is then:
ηkρIm=k=0kηkqIm
The corresponding transfer function of Eq. (28) with a pole at z = 1 has a low-pass response singular at DC. In general lower frequency components are amplified more strongly via the corresponding transfer function
HQ(ejω)=z/(z1)|z=ejω=j2ejω/2/sin(ω/2)
Thus, the Q components of the input noise (associated with the phase walkoff) coincide with the independent increments of a random walk process, Eq. (30), for the phase of the field (enhancing the Wiener random walk already present due to the laser phase noise). This FR processing induced phase noise process has the same general statistics as the laser phase noise, thus may be partially mitigated by the carrier recovery system. However, the amplitude noise in the reconstructed field poses more of a problem. We may express the relative amplitude variation ηkqRe in Eq. (29) explicitly in terms of the input noise sequence, by determining the impulse response of the corresponding recursion ηkρRe=ηkqReηk1ρReof Eq. (26), to which we refer to as alternating accumulator (ALT-ACC), as its impulse response is given by the sign alternating sequence (1)kuk. The ALT-ACC system output is then expressed as a convolution with the following impulse response:
ηkρRe=k=0k(1)kkηkqRe=(1)kk=0k(1)kηkqRe
This is also a running sum of the input noise samples up to time k, except that the signs are alternated prior to the noise elements ηkqRebeing summed up. It is apparent that low frequency components of the input noise will be strongly attenuated by the ALT-ACC transfer function for the amplitude (I component), whereas noise components in the vicinity of half the sampling frequency tend to be strongly amplified. In particular a noise component of the form ejπk=(1)k, at precisely half the sampling frequency, corresponding to an angular frequency on the unit circle right at the pole z = −1, yields a divergent running sum, i.e. the output amplitude noise due to this components grows without bound. The corresponding high-pass transfer function, with singularity at ω=πis obtained by either exciting the system with a sequence ejω, or evaluating the Z-transform over the unit circle:
HI(ejω)=(1)kk=0k(1)kejω=z/(z+1)|z=ejω=12ejω/2/cos(ω/2)
As the samples ηkqReare uncorrelated for different k, and so are the samples, ηkqIm, then the running sums from 0 to k occurring in the last expression are readily shown to be cumulative random processes with monotonically increasing variance. Inspecting Eqs. (30) and (32), it is apparent both the magnitude and phase of the reconstructed field ρ˜^ktend to “run-off”, i.e. accumulate along a random walk, though the selectivities of the magnitude and phase impairments with respect to noise frequency components are different. As long as the output noise remains small, the reconstructed field is given by following expression, derived from the approximation of Eq. (29)):
ρ˜^k=ρ˜^ko(1+ηkqRe+jηkqIm)ρ˜^ko(1+(1)kk=0k(1)kηkqRe)exp{jk=0kηkqIm}
In this expression the amplitude (phase) run-off is entirely determined by accumulation of the I (Q) components of the noise in the FR module input, respectively (for the I-components the accumulation is with alternating sign). Notice that the I and Q noises η˜kqRe,η˜kqImare each uncorrelated processes but are not stationary (hence are not white), as the original nkqRe,nkqIm noises before normalization are stationary and white, but the normalization renders them non-stationary. Assuming for a moment that the processes η˜kqRe,η˜kqImare stationary (e.g. if the received field ρ˜kis constant, thus we normalize by a constant), then the two processes η˜kqRe,η˜kqImwould be white and would have frequency components over the full spectral band up from DC up to the sampling frequency (from 0 to 2π) and in particular they would have frequency components in the vicinity of DC and half the sampling frequency (0 and π), and these frequency components would be strongly amplified, thus the both the reconstructed amplitude and phase will run without bound. The amplitude and phase random walks constitute two impairment mechanisms degrading the proposed FR procedure:(i): The reconstructed field is affected by a phase-noise mechanism, and the amplitude also tends to run-off as an independent increments random walk process. (ii): It appears that the FR recursion may only be run for a finite number of steps, following initialization, before the numerical realization of the FR module tends to run into overflow or underflow and must be reset. Despite these seemingly insurmountable impairments, in the sequel we propose a functioning SC receiver structure, which may even outperform coherent detection in a certain operational regime.

5. Twice-oversampled self-coherent single polarization receiver

This paper is devoted to establishing the principles of field reconstruction for just a single polarization Rx. The extension to a dual-polarization Rx is deferred to a future publication, while here we treat just single polarization operation, to facilitate the assimilation of the new concepts.

The analysis of the FR operation in section 3 indicates that a key issue to contend with is the magnitude scaling alternation effect affecting the even and odd polyphase components generated in the field reconstruction process. To overcome this impairment we propose to use a twice-oversampled receiver, i.e. A/D convert the received signal with two samples per symbol. Further downstream in the processing chain, prior to carrier recovery and decision, the sampling rate is to be halved back down to the baud-rate. This implies that it is either the even or odd polyphase that is extracted and retained to be baud-rate processed in the carrier recovery stage. Thus, a common gain factor is experienced by all the samples of twice-down-sampled signal; no longer do we need to contend with differing gains for the even and odd polyphase subsequences, as a single polyphase is retained. Although the common gain factor affecting the baud-rate signal is unknown, this scaling factor is automatically calibrated by the AGC-like capability of the adaptive MSDD carrier recovery (CR) module. Similarly, the unknown but fixed phase offset affecting the field-reconstructed sequence is also de-rotated away by the MSDD CR, compensating for any tilt of the detected QAM constellation.

Figure 3 describes a complete scalar (single polarization) link, including the transmitter (a), the optical channel (b), and two alternative receiver structures, namely self-coherent (c) and coherent (d) ones. The transmitter model described in Fig. 3(a) features a single-carrier 16-QAM mapper feeding a modulus-preserving differential precoder as described in [8], where A˜k are the line symbols and s˜k are the complex-valued information symbols, selected out of the 16-QAM constellation:

A˜k=s˜kA˜k1/|A˜k1|
These differentially precoded line symbols are next up-sampled by a factor of 2, then fed into a raised cosine (RC) filter with roll-off factor of 0.01 and group delay of 1024 taps and band-limitation equal to one-half of the sampling rate. This filter acts as interpolation and shaping filter, also emulating the DAC action of the transmitter.

 

Fig. 3 Scalar (single-polarization) link. (a): Single channel 16-QAM transmitter with modulus preserving differential precoding [8] (b): Simple additive white Gaussian (ASE) noise and laser phase noise channel model. (c): SC single-polarization receiver using and DIs with half baud-interval delay and 2x oversampled ADC. (d): Comparable fully-coherent receiver. For coherent detection the channel model is modified by doubling the effective linewidth (making the substitution Δν2Δν) to account for the combined effect of transmit laser and OLO laser.

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The simple non-dispersive channel model (Fig. 3(b)) adopted here for testing robustness to phase noise, is as described in [8], accounting just for ASE induced additive white Gaussian noise, n˜kw and laser phase noise, which is expressed as a Wiener-Levy random walk process, ϕk=m=0kΩm+ϕ0;Ωm~N[0,2πΔν12T], where Δν is the laser linewidth, and T is the symbol interval (the factor of ½ is due to the 2-fold oversampling).

Comparing the SC Rx (Fig. 3(c)) and the fully-coherent Rx (Fig. 3(d)), these two structures differ in the presence/absence of the FR module, and also differ in having a DI-based vs. an optical local oscillator based Rx front-end. The complex field estimate generated in the SC Rx may be further processed just as in a conventional coherent receiver, in order to mitigate optical channel impairments such as CD, polarization mixing and phase noise. Here we just model a scalar (single polarization) receiver over a short-haul link with negligible CD, hence the only post-FR module included in the post-processing chain is the MSDD CR.

Nevertheless, merely twice oversampling followed by twice down-sampling is still not sufficient. It turns out that we must also replace our original DIs which have delay T (equal to the symbol interval duration) with new DIs which have fractional delay T/2 (the feasibility of using fractional symbol rate delays for obtaining multiple samples per symbol of the reconstructed field was previously established in [3] [4]). Thus, we propose to adopt DIswith fractional delay of half the symbol duration, and sample their outputs in the ADC at twice the symbol rate. The SC Rx front-end (Fig. 3(c)) comprises an IQ DI with a half-symbol interval delay and an ADC clocked at twice the baud-rate. To analyze the impact of the half-symbol delay in the DIs, we recall Eqs. (3), (4) stating that the DI samples q˜kρ˜kρ˜k1* are obtained by sampling the DI analog outputs at intervals t=kτDI, and denoting the respective samples by q˜k=q˜(kτDI)and ρ˜k=ρ˜(kτDI). In our specific case, these equations hold with τDI=T/2. In particular, the FR algorithm ρ˜^k=q˜k/ρ˜^k1*operates ‘as usual’, just with the discrete time k interpreted now as running at twice the symbol rate, i.e., we obtain two samples of reconstructed field per symbol, rather than one sample as before.

An ancillary benefit of using DIs at half the fractional delay, is that these DIs are easier to implement and are more robust, as the DI optical path delays get shortened by a factor of two. Ancillary benefits the twice-oversampling are that the FR may be followed by an oversampled single-carrier receiver, which is more advantageous than a baud-rate (one-symbol-per-sample) Rx, and not too hard to implement at the low baud-rates. A key advantage of the fractional sampling receiver is simplified timing recovery (TR). The simplest TR approach is to select one of the two even and odd polyphases for which the eye is most open. A more advanced TR solution is to use timing interpolation techniques [12], however the issue of TR techniques for SC detection will not be further pursued here – the simulations will assume ideal timing.

For carrier recovery use a variant of the MSDD (Fig. 4 ) which incorporates an adaptive NLMS AGC. In contrast, our recent MSDD CR system for QAM [9] was based on a Least Mean Squares (LMS) scheme, lacking the normalization featuring in NLMS, which turns out to be essential for SC field reconstruction. The NLMS MSDD adaptive module may be switched from a data-aided (training sequence driven) mode to a decision-directed (decision feedback driven) mode.

 

Fig. 4 Multi-Symbol Delay Detection (MSDD) Carrier Recovery system, incorporating a modified adaptive NLMS AGC algorithm. This system resembles the U-notU MSDD disclosed in [9] but differs from it in using a more rapidly converging Normalized LMS (NLM) rather than an LMS MSDD algorithm.

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Our simulations indicate that it suffices to initialize the system with a training sequence, then permanently switch to the decision-directed mode for the whole remaining duration of the transmission.

The main detrimental effect of RF output noise accumulation is the amplitude noise run-off (as the phase run-off is partially addressed by the carrier recovery stage as it amounts to an extra source of effective laser phase noise. Moreover, in in terms of its relative magnitude, this effective phase noise source is weaker than the laser phase noise in the low baud-rate transmission regime of interest. The good news is that the amplitude random wander is relatively slow, thus providing good-quality AGC functionality in the processing stages following the FR, should be able, in principle, to take out the amplitude wander. This is the essence of our AGC-based amplitude noise mitigation approach. In the SC Rx of Fig. 3(c) the AGC function is embedded in the NLMS MSDD CR module.

In the next section we show by simulation that in the absence of an AGC, a stringently low level of quantization noise would be required, calling for a 14-bit ADC. In section 8 we show that upon incorporating the MSDD module of Fig. 4 into the SC Rx chain, which provides the AGC function, results in reducing the ADC requirements down to 9-11 bits.

6. Numerical simulations of quantization noise random walk at the FR output

In this section we numerically explore the noise properties of the field reconstruction procedure, verifying the analytical noise model of section 4. Here we perform a Tx-Rx back-to-back simulation accounting for the Rx ADC effect, monitoring the output of the FR module prior to having it further propagate it through the MSDD CR (that final step will be pursued in section 7).

The SC Rx front-end considered in this section (Fig. 5 ) comprises an IQ DI with a half-symbol interval delay and an ADC clocked at twice the baud-rate, featuring a variable number, B, of ADC bits – the case of negligible quantization noise is also tested by setting B = 32. The ADC feeds the FR module which is in turn followed by an adaptive complex-valued AGC (C-AGC) module activated in conjunction with a training sequence, which is periodically activated and de-activated. The C-AGC is not just a conventional magnitude-only AGC, but it rather optimizes a complex-valued scaling parameter including gain and phase. The C-AGC realizes a single-complex-valued tap NLMS algorithm. We note that this C-AGC module is used in this section solely for the purpose of initializing meaningful field reconstruction error metrics, but is not part of our final version of SC Rx.

 

Fig. 5 Block diagram of simulated field reconstruction post-detection noise impairment and its mitigation through oversampling and bandlimiting. (a): Transmitter model. (b): Receiver model. The switch SW has two positions corresponding to either using the band-limiting filter or not using it.

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Whenever the training sequence is de-activated, the C-AGC is also de-activated, in effect bypassed. Such intervals are referred to as ‘free-run’, whereas the intervals during which the training sequence is activated and the C-AGC is locked are referred to as the data-aided (DA) intervals. The role of the C-AGC is to reset the system to nearly the correct initial condition at the beginning of each ‘free-run’ interval. To this end, the complex gain coefficient is frozen at the end of the DA interval, once the receiver starts detecting transmitted info symbols. The C-AGC then initializes the errors Eq. (37) to near zero at the beginning of each detection interval. The actual SC Rx presented in the next section does not require a separate C-AGC module, as the equivalent gain control (AGC) and phase offset de-rotation functions are effectively carried out by the MSDD carrier recovery system, providing even better performance.

Our simulations in this section account for the time-evolution of ADC noise accumulation at the FR output, comparing SC Rx front-end versions differing in the absence or presence of quantization noise, varying the ADC bit counts, B, when the quantization noise is evaluated. In order to monitor the amplitude and phase walk-off we repeatedly reset the system during each DA interval, then let the system evolve by itself over the following ‘free-run’ interval, obtaining the time traces of Fig. 6 for the FR magnitude and phase errors for various ADC bit counts. Remarkably, when using B = 32 bits (negligible quantization noise), the resulting FR magnitude and phase errors come out practically zero, i.e., the FR algorithm outlined in sub-section 2.3 works perfectly, up to a fixed gain and phase error, which is taken out by the C-AGC, thus the resulting error errors are indeed null. This has been numerically simulated but the errors coincide so well with the zero axis, that it is not possible to display a distinct curve.

 

Fig. 6 Magnitude and phase errors time evolution for the SC receiver front end of Fig. 5, plotted over ten cycles of 20 training symbols with C-AGC tracking, followed by 180 info symbols, with the C-AGC frozen to the last value attained at the end of the preceding training sequence. The yellow square wave is an indicator of the data-aided mode – it is “on” during the 20 symbols long training sequences, while it is “off” during the 180 symbols long operating intervals during which the FR module reconstructs the received field. (a,b): Respective relative magnitude and phase errors for B = 12 bits ADC. (c,d): Respective relative magnitude and phase errors for B = 13 bits ADC. (e,f): Respective relative magnitude and phase errors for B = 14 bits ADC. In all phase error plots, (b,d,f), the “wildest” curve (red) is the total received phase noise of a reference coherent system, due to the combined effect of the transmitter and receiver OLO lasers. It is apparent that the phase wander induced by the ADC noise is substantially smaller than that due to the laser phase noise.

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The performance metrics used here to describe the field reconstruction fidelity in the presence of quantization noise are the FR magnitude and phase errors, defined as deviations between the reconstructed (FR output) and actual field (input into DI) magnitude and phase:

εkρ|ρ˜^k||ρ˜k|;εkϕρ˜^kρ˜k
Once the quantization noise impairment is included (setting the ADC bit count B to 12...14) a random walk clearly emerges in magnitude and phase, starting at time indexes whereat the C-AGC, complex scaling factor is frozen to the last value it had during the initial 20-symbols data-aided training interval. The magnitude and phase run-offs become successively worse as the ADC bit count is reduced. For B = 14 bits, the relative magnitude error wander is limited to a 5% band, whereas the phase error is limited to a several tens of mili-radians, corresponding to a not-too-excessive overall impairment, enabling in principle SC detection of a 16-QAM transmitted signal. When plotting the phase errors we have also superposed, for comparison, the phase noise wander of a coherent system using 100 KHz linewidth lasers in the Tx and the Rx OLO. It is apparent that the phase noise induced by ADC quantization is substantially smaller than the laser phase noise present in a coherent Rx. An important observation is that the SC Rx is affected by laser phase noise to a smaller extent, due to the lack of the local oscillator laser in the Rx. The laser phase noise increments variance in the SC Rx is then just half that of a conventional coherent system, wherein the transmit and receive equal contributions to laser phase noise are combined (assuming identical lasers for the transmitter and the OLO). The good news is that the doubling of the laser phase noise tolerance in the SC system relative to a coherent one is not offset by a substantial increase in numerical phase noise error induced by the ADC quantization accumulating through the FR divisor. Indeed, Fig. 6 indicates that numerical phase noise induced in the FR process appears to be up an order of magnitude less intense than the laser phase noise assuming 100 KHz linewidth lasers and 100 MBd baud-rate.

The main remaining concern for SC detection is the cumulative run-off in the magnitude error. Figure 6 indicates that it takes as many as 14 bits in the ADC to keep this magnitude error in check over hundreds of symbols, once the training sequence is turned off. A 14 bits ADC would technologically imply a severe limitation on the baud-rate of SC detection, since large ADC bit counts are quite hard to realize at higher sampling rates. Fortunately, we shall show in the next section that incorporating a decision-directed AGC in the MSDD carrier recovery stage substantially eases up the amplitude resolution requirement of the AGC, taking it down to 9-11 bits.

7. Numerical simulations of scalar (single pol.) SC 16-QAM transmission

In this section we present simulations of the performance of the complete single-polarization SC Rx, including the adaptive NLMS MSDD CR module, which provides both phase noise mitigation but also the critical on-line AGC capability used during active transmission (not to be confused with the C-AGC artificial functionality of the previous section).

The relative performances of the coherent and self-coherent 16-QAM single polarization receivers of Fig. 3(c,d) are compared in Fig. 7 , which plots BER vs. OSNR assuming a variety of parameters, varying baud-rate, laser linewidth, and the number of bits in the ADC. It turns out that the resilience of the overall SC Rx chain is significantly improved relative to that implied in the simulations of the previous section. The reason for this is the beneficial effect of having the on-line AGC capability, as provided by the MSDD. We also remark that in the SC Rx, the ASE noise has a very different impact than ADC quantization noise does. We have seen in section 4 that ADC quantization noise builds up cumulatively through the recursive divider. In contrast, the ASE noise is just a part of the composite noisy signal at the DI input, thus is linearly reproduced at the FR output, as the FR in effect acts as an inverse DI, undoing the DI non-linear transformation applied to its input field.

 

Fig. 7 Bit Error Ratio (BER) vs. Optical Signal to Noise Ratio (OSNR) performance of a scalar (single) polarization SC Rx with optical amplification, vs. a fully coherent receiver, for 0 KHz, 100 KHz and 200 KHz laser linewidth (LW) and for 100 MBd and 200 MBd baud rates. (top to bottom): various numbers of bits in the ADC, as listed in the header of each plot.

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8. Divisive exception - division by zero resulting from extremely low field values

Finally, we note that the FR divider accuracy is degraded whenever its input value becomes too low or zero, an event referred to as divisive exception. In the extreme case when the digit-ized ADC output is zero (i.e., whenever the I-DI or Q-DI output is less than half an LBS in magnitude), a divide-by-zero catastrophic exception occurs in the next discrete-time interval, once the zero value loops back to the divisor input.

To reduce outages, the mean time between underflows/divide-by-zero-exceptions must be made as large as possible. When chromatic dispersion in the optical channel is moderate or low and as long as the signal to noise ratio is not too poor (i.e. the probability of very negative noise peaks is still very low), and further assuming that the received field is sampled at the point where the eye is most open, the occurrences of very low field values, hence of divisive exception are quite rare. The low CD requirement re-enforces the conclusion that the most suitable applications for SC field reconstruction are indeed in the metro/access domain rather than long-haul transmission. This discussion also indicates that SC detection would not function very well with modulation formats with large PAPR, such as OFDM, but rather single-carrier formats are preferably used with FR based self-coherent detection.

A different strategy to prevent the divisive-exception is to convert the null field value into the FR into a very small, yet non-zero value, the division by which would correspond to a high value, which is nevertheless finite. In our simulation this was achieved by the following modification to the basic FR algorithm of Fig. 2, as described in Fig. 8 . The idea is to additively inject into the divisive loop a small constant ε0or a very low power random sequence εk. If the sample of q˜k0 goes null (which occurs whenever 12LSB<q˜(k0τDI)<12LSB), then the division yields zero, i.e. ρ˜^k0=0. After a unit delay, at time k0+1, the lower port of the divisor would become zero, yielding a divide-by-zero exception. However, the addition of εkmakes the divisor value εk0+ρ˜^k0 practically non-zero (it is extremely improbable that ρ˜^k+εk hit precisely zero), eliminating the divisive exception. This methodology has practically proven itself in removing divisive exception over the course of simulations, however further study is warranted to determine its residual outage statistics, its impact on performance, and further improvements, detecting the occurrence of zero and slightly shifting it.

 

Fig. 8 Field reconstruction algorithm with evasion of the divisive-exception.

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9. Perspective and concluding remarks

Beyond containing concluding remarks, this section presents important perspective and elucidates some subtle points concerning the comparison of SC vs. fully-coherent detection.

Our main accomplishments re SC detection in this paper consist of the following:

A. Our proposed novel FR algorithm satisfactorily performs joint reconstruction of both amplitude and phase at once, in a relatively accurate manner, just based on the IQ DI outputs (no IM branch required), by directly operating in the complex field envelope domain. This FR algorithm would yield infinite precision in the ideally noiseless case, whereas prior magnitude reconstruction mechanisms just based on the DI outputs would generate gross errors even in the ideally noiseless case.

B. Analytical analysis of amplitude and phase noise build-up in the field reconstruction algorithms, verified by thorough numeric simulation.

C. In addition to numerically induced extra phase noise, the SC Rx must mainly contend with magnitude errors, which were identified as the main concern. To overcome the two impairments of reconstructed field magnitude even-odd samples alternation and the FR noise buildup (manifesting as random walk run-offs due to the ADC noise accumulation in the FR divider), we introduced a novel SC Rx architecture comprising DIs with fractional (half-symbol-rate) delay, twice oversampling and a novel carrier recovery module based on MSDD decision-directed adaptive NLMS AGC capability.

The identified deficiencies or weaker points of SC detection are:a. Numerical precision requirements: Even upon applying our amplitude noise mitigation technique (without which the numerical errors would be overwhelming), the numerical precision requirements remain substantially higher for SC detection than for conventional coherent detection – which is the main price to pay for the elimination of the local oscillator laser and its replacement by the advanced field reconstructing DSP. Nevertheless, the good news is once we invest the extra amplitude resolution, providing 9-11 bits in the ADC, the magnitude noise random walk is well tracked and mitigated and the SC Rx advantage in phase noise mitigation shows up as discussed below, such that the SC Rx outperforms the coherent Rx.b. Dynamic range issues: potential outage due to the divisive exception, as discussed in section 8. Thus, occurrence of very low or zero values in the optical field at the SC Rx input should be avoided, which might limit the range of applicability of SC detection to systems wherein the chromatic dispersion is not significant. Moreover, a more complex timing mechanism might be necessary in the SC receiver in order to sample the eye at the point where it is most open, thus avoiding the very low field values. However, in the last section we introduced a divisive exception mitigation strategy which was empirically shown to work in our simulations, but the whole issue is subject to additional research.

There are multiple technical and economic factors leading to the conclusion that SC detection, in the form presented in this paper, is most suitable for and in fact restricted to, transmission systems at ‘modest’ symbol rates of hundreds-of-MBd, as applicable in particular to the next generation of PON optical access systems:i. LO elimination: An economic driver is that conventional coherent detection requires a more complex optical receiver front-end (FE) comprising an optical local oscillator, hence its main applicability is for long-haul optical links. However, the cost and power consumption of the OLO laser are still prohibitive for applications which are highly sensitive to receiver opto-electronic hardware complexity, such as metro networking and especially high-speed optical access based on next generation PON. In particular, as PON system target lower reach and require lower cost Optical Network Units (ONU), the DI-based DDD systems would seem to make economic sense in terms of cost for this access application, relative to using the more complex coherent systems. Unfortunately, the DDD capacity performance is not satisfactory, as higher order modulation formats and polarization multiplexing are currently not supported with DDD transmission. The requirement for higher-speed upgrades in access networks indicates that it would be desirable to find a way to adapt DDD to attain the well-known advantages of coherent detection, albeit without incurring the cost and complexity of an OLO in the ONU receivers, which must be kept at low cost. In this respect we mention that the SC front-end of Fig. 1(b) uses the same hybrid as in coherent detection, thus the elimination of the OLO laser is not offset by additional optical front-end complexity.ii. Enhanced phase noise tolerance: The SC Rx is approximately twice as phase-noise tolerant as the coherent Rx is, however this advantage is most meaningfully manifested at low baud-rates. The absence of the OLO laser in the SC receiver approximately halves the laser phase noise (assuming the Tx laser and Rx laser have identical linewidths), however an artificial new source of phase noise is introduced, namely the numerically generated cumulative phase noise at the output of the FR module. Fortunately, this phase noise source turns out to be relatively weak relative to the laser phase noise, for transmission systems at ‘modest’ symbol rates of hundreds-of-MBd, as applicable to PON optical access systems. Indeed, at low baud rates, whereat the SC systems are constrained to operate, the accumulation of laser phase noise is enhanced due to the long duration of the low-baud rate symbols, over which the laser phase noise random walk accumulates more variance. Therefore, it is particularly in this low baud-rate regime that halving the laser phase noise would provide significant savings, with respect to which the enhanced numerical phase noise at the FR output due to the ADC quantization noise, would be relatively small. Therefore, for low baud-rate applications such as PON, SC Rx is, to a good approximation, twice as tolerant of laser phase-noise than a fully-coherent one is. For PON systems this factor-of-two relief yields a significant performance advantage, as low-baud-rate fully coherent systems are severely degraded by phase noise. In this respect, we note that the transition to next-generation PON does not necessarily enhance the baud-rate. Rather the bit-rate is projected to be enhanced by roughly maintaining the same baud-rate but increasing spectral efficiency by using higher-order QAM modulation formats.iii. Extra ADC resolution: SC detection requires a higher number of effective bits in the ADC, which is a major deficiency, but the enhanced ADC requirements are more readily met at lower sampling rates. In particular at sub-GS/s sampling rates it is possible to attain the required number of effective bits. Specifically, a SC receiver capable of supporting 16-QAM detection at the baud rate of 100MBd, 200MBd was shown to require an ADC with as many as 10, 11 bits in order to provide sufficient precision to the field magnitude reconstruction process, but then such SC Rx outperforms coherent detection due to the elimination of the OLO laser phase noise. This indicates that, practically, given the rate of progress in ADC technology, the targeted baud-rate should be of the order of 200 MBd, corresponding to 400 MS/s after 2-fold oversampling, at which sampling rate 11 bits ADC are currently available. Notice that 200 MBd transmission may carry ~1.2 Gb/s payload over polarization multiplexed 16-QAM modulation, including overheads, which may be adequate for next generation PON.iv. ASE beat-limited regime: SC detection requires an adequate amount of optical pre-amplification such that the receiver become ASE-beat noise limited, i.e. the ASE noise dominate over the Rx thermal noise, which condition is most readily achieved for low-bandwidth applications, as the receiver noise equivalent current tends increase for wider-band optical receivers. Still, this condition might mean a reduction in the reach of the PON link.

Related to this issue, let us make the important observation is that SC detection is impaired by the presence of post-DI noise, however, the pre-DI noise added in the optical channel, such as optical amplifier noise, may be considered as an integral part of the input optical field, which is quite precisely reconstructed in the FR, which generates a coherent estimate of the input field including its accompanying noise. Thus, the ASE pre-DI noise is linearly reproduced at the FR output, as the FR in effect acts as an inverse DI, undoing the DI non-linear transformation applied to its input field. It is then up to the processing stages following the FR, to mitigate the optical channel noise, just as DSP stages in a fully coherent receiver would do. In contrast, post-DI noise sources, such as the ADC quantization noise, and the thermal receiver noise are detrimental to the quality of field reconstruction itself. In this paper we heretofore assumed that the system is ASE-beat-noise limited, i.e. sufficient optical gain is provided such that the thermal noise in the optical receivers is negligible relative to the ASE noise and may thus be neglected relative to the ADC quantization noise, which was the sole degradation effect considered to impact the FR estimate quality. If there were residual thermal noise, then its effect would be akin to increasing quantization noise, thus the effective number of bits (ENOB) of the ADC would need to be reduced, which would degrade the FR fidelity. However, as we assumed ASE beat-noise-limited operation, the only effective source of noise at the digitized IQ DI outputs would be ADC quantization noise, as assumed in the simulation.

Finally future work will extend the SC operation from single to dual polarization. Additional imperfections which would certainly impact the SC receiver performance are IQ imbalances of the DIs, and numerical precision requirements in the FR hardware (as opposed to the ADC). These issues will also be treated in a future publication.

Appendix - Glossary

ADC = Analog to Digital Converter
DP = differential precode
NLMS = Normalized Least Mean Squares
AGC = Automatic Gain Control
FR = field reconstruction
OLO = Optical Local Oscillator
C-AGC = Complex AGC
IM-DD = Intensity Modulated Direct Detection
PON = Passive Optical Network
CR = carrier recovery
LMS = Least Mean Squares
Rx = Receiver
DDD = Differential Direct Detection
LTI = linear time-invariant
SC = Self Coherent
DBPSK/DQPSK = Differential 
Binary/Quaternary Shift Keying
MSDD = Multi Symbol Delay Detection
Tx = Transmitter
DI = Delay Interferometer

Acknowledgments

This work was supported in part by the Israeli Science Foundation (ISF), by the OTONES trans-national Piano + EU program, and by the EURO-FOS Network of Excellence project.

References and links

1. N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and First Experimental Demonstration of Digital Incoherent Optical Field Detector for Chromatic Dispersion Compensation,” in ECOC’05 European Conference of Optical Communication, PDP Th. 4.4.4 (2005).

2. J. Zhao, M. E. McCarthy, and A. D. Ellis, “Electronic dispersion compensation using full optical-field reconstruction in 10Gbit/s OOK based systems,” Opt. Express 16(20), 15353–15365 (2008). [CrossRef]   [PubMed]  

3. X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16(2), 792–803 (2008). [CrossRef]   [PubMed]  

4. N. Kikuchi and S. Sasaki, “Highly Sensitive Optical Multilevel Transmission of Arbitrary Quadrature-Amplitude Modulation (QAM) Signals With Direct Detection,” J. Lightwave Technol. 28(1), 123–130 (2010). [CrossRef]  

5. Y. Takushima, H. Y. Choi, and Y. C. Chung, “Transmission of 108-Gb/s PDM 16ADPSK signal on 25-GHz grid using non-coherent receivers,” Opt. Express 17(16), 13458–13466 (2009). [CrossRef]   [PubMed]  

6. J. Li, R. Schmogrow, D. Hillerkuss, M. Lauermann, M. Winter, K. Worms, C. Schubert, C. Koos, W. Freude, and J. Leuthold, “Self-Coherent Receiver for PolMUX Coherent Signals, ” in OFC/NFOEC’11 Conference on Optical Fiber Communication, OWV5 (2011).

7. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications (Springer, 2011).

8. N. Sigron, I. Tselniker, and M. Nazarathy, “Carrier phase estimation for optically coherent QPSK based on Wiener-optimal and adaptive Multi-Symbol Delay Detection (MSDD),” Opt. Express 20(3), 1981–2003 (2012). [CrossRef]   [PubMed]  

9. I. Tselniker, N. Sigron, and M. Nazarathy, “Joint phase noise and frequency offset estimation and mitigation for optically coherent QAM based on adaptive multi-symbol delay detection (MSDD),” Opt. Express 20(10), 10944–10962 (2012). [CrossRef]   [PubMed]  

10. M. Nazarathy, Y. Yadin, M. Orenstein, Y. K. Lize, L. Christen, and A. E. Willner, “Enhanced Self-Coherent Optical Decision-Feedback-Aided Detection of Multi-Symbol M-DPSK/PolSK in particular 8-DPSK/BPolSK at 40 Gbps,” in OFC/NFOEC’07 Conference on Optical Fiber Communication (2007).

11. S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-Aided Carrier Phase Estimation for Coherent Optical Communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]  

12. H. Sun and W. K. Tsan, “Clock recovery and jitter sources in coherent transmission, paper OTh4C.2,” in OFC/NFOEC’ Conference on Optical Fiber Communication, OTh4C.2 (2012).

References

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  1. N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and First Experimental Demonstration of Digital Incoherent Optical Field Detector for Chromatic Dispersion Compensation,” in ECOC’05 European Conference of Optical Communication, PDP Th. 4.4.4 (2005).
  2. J. Zhao, M. E. McCarthy, and A. D. Ellis, “Electronic dispersion compensation using full optical-field reconstruction in 10Gbit/s OOK based systems,” Opt. Express 16(20), 15353–15365 (2008).
    [CrossRef] [PubMed]
  3. X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16(2), 792–803 (2008).
    [CrossRef] [PubMed]
  4. N. Kikuchi and S. Sasaki, “Highly Sensitive Optical Multilevel Transmission of Arbitrary Quadrature-Amplitude Modulation (QAM) Signals With Direct Detection,” J. Lightwave Technol. 28(1), 123–130 (2010).
    [CrossRef]
  5. Y. Takushima, H. Y. Choi, and Y. C. Chung, “Transmission of 108-Gb/s PDM 16ADPSK signal on 25-GHz grid using non-coherent receivers,” Opt. Express 17(16), 13458–13466 (2009).
    [CrossRef] [PubMed]
  6. J. Li, R. Schmogrow, D. Hillerkuss, M. Lauermann, M. Winter, K. Worms, C. Schubert, C. Koos, W. Freude, and J. Leuthold, “Self-Coherent Receiver for PolMUX Coherent Signals, ” in OFC/NFOEC’11 Conference on Optical Fiber Communication, OWV5 (2011).
  7. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications (Springer, 2011).
  8. N. Sigron, I. Tselniker, and M. Nazarathy, “Carrier phase estimation for optically coherent QPSK based on Wiener-optimal and adaptive Multi-Symbol Delay Detection (MSDD),” Opt. Express 20(3), 1981–2003 (2012).
    [CrossRef] [PubMed]
  9. I. Tselniker, N. Sigron, and M. Nazarathy, “Joint phase noise and frequency offset estimation and mitigation for optically coherent QAM based on adaptive multi-symbol delay detection (MSDD),” Opt. Express 20(10), 10944–10962 (2012).
    [CrossRef] [PubMed]
  10. M. Nazarathy, Y. Yadin, M. Orenstein, Y. K. Lize, L. Christen, and A. E. Willner, “Enhanced Self-Coherent Optical Decision-Feedback-Aided Detection of Multi-Symbol M-DPSK/PolSK in particular 8-DPSK/BPolSK at 40 Gbps,” in OFC/NFOEC’07 Conference on Optical Fiber Communication (2007).
  11. S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-Aided Carrier Phase Estimation for Coherent Optical Communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010).
    [CrossRef]
  12. H. Sun and W. K. Tsan, “Clock recovery and jitter sources in coherent transmission, paper OTh4C.2,” in OFC/NFOEC’ Conference on Optical Fiber Communication, OTh4C.2 (2012).

2012 (2)

2010 (2)

2009 (1)

2008 (2)

Chandrasekhar, S.

Chen, J.

Choi, H. Y.

Chung, Y. C.

Ellis, A. D.

Kam, P. Y.

Kikuchi, N.

Leven, A.

Liu, X.

McCarthy, M. E.

Nazarathy, M.

Sasaki, S.

Sigron, N.

Takushima, Y.

Tselniker, I.

Yu, C.

Zhang, S.

Zhao, J.

J. Lightwave Technol. (2)

Opt. Express (5)

Other (5)

M. Nazarathy, Y. Yadin, M. Orenstein, Y. K. Lize, L. Christen, and A. E. Willner, “Enhanced Self-Coherent Optical Decision-Feedback-Aided Detection of Multi-Symbol M-DPSK/PolSK in particular 8-DPSK/BPolSK at 40 Gbps,” in OFC/NFOEC’07 Conference on Optical Fiber Communication (2007).

J. Li, R. Schmogrow, D. Hillerkuss, M. Lauermann, M. Winter, K. Worms, C. Schubert, C. Koos, W. Freude, and J. Leuthold, “Self-Coherent Receiver for PolMUX Coherent Signals, ” in OFC/NFOEC’11 Conference on Optical Fiber Communication, OWV5 (2011).

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications (Springer, 2011).

H. Sun and W. K. Tsan, “Clock recovery and jitter sources in coherent transmission, paper OTh4C.2,” in OFC/NFOEC’ Conference on Optical Fiber Communication, OTh4C.2 (2012).

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and First Experimental Demonstration of Digital Incoherent Optical Field Detector for Chromatic Dispersion Compensation,” in ECOC’05 European Conference of Optical Communication, PDP Th. 4.4.4 (2005).

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Figures (8)

Fig. 1
Fig. 1

SC receiver front-end alternatives. (a): An IQ DI realization consisting of a pair of delay interferometers in quadrature. (b): An equivalent 90 deg hybrid-based realization of the IQ DI.

Fig. 2
Fig. 2

The field reconstruction algorithm embedded in a schematic of a self-coherent receiver.

Fig. 3
Fig. 3

Scalar (single-polarization) link. (a): Single channel 16-QAM transmitter with modulus preserving differential precoding [8] (b): Simple additive white Gaussian (ASE) noise and laser phase noise channel model. (c): SC single-polarization receiver using and DIs with half baud-interval delay and 2x oversampled ADC. (d): Comparable fully-coherent receiver. For coherent detection the channel model is modified by doubling the effective linewidth (making the substitution Δν2Δν ) to account for the combined effect of transmit laser and OLO laser.

Fig. 4
Fig. 4

Multi-Symbol Delay Detection (MSDD) Carrier Recovery system, incorporating a modified adaptive NLMS AGC algorithm. This system resembles the U-notU MSDD disclosed in [9] but differs from it in using a more rapidly converging Normalized LMS (NLM) rather than an LMS MSDD algorithm.

Fig. 5
Fig. 5

Block diagram of simulated field reconstruction post-detection noise impairment and its mitigation through oversampling and bandlimiting. (a): Transmitter model. (b): Receiver model. The switch SW has two positions corresponding to either using the band-limiting filter or not using it.

Fig. 6
Fig. 6

Magnitude and phase errors time evolution for the SC receiver front end of Fig. 5, plotted over ten cycles of 20 training symbols with C-AGC tracking, followed by 180 info symbols, with the C-AGC frozen to the last value attained at the end of the preceding training sequence. The yellow square wave is an indicator of the data-aided mode – it is “on” during the 20 symbols long training sequences, while it is “off” during the 180 symbols long operating intervals during which the FR module reconstructs the received field. (a,b): Respective relative magnitude and phase errors for B = 12 bits ADC. (c,d): Respective relative magnitude and phase errors for B = 13 bits ADC. (e,f): Respective relative magnitude and phase errors for B = 14 bits ADC. In all phase error plots, (b,d,f), the “wildest” curve (red) is the total received phase noise of a reference coherent system, due to the combined effect of the transmitter and receiver OLO lasers. It is apparent that the phase wander induced by the ADC noise is substantially smaller than that due to the laser phase noise.

Fig. 7
Fig. 7

Bit Error Ratio (BER) vs. Optical Signal to Noise Ratio (OSNR) performance of a scalar (single) polarization SC Rx with optical amplification, vs. a fully coherent receiver, for 0 KHz, 100 KHz and 200 KHz laser linewidth (LW) and for 100 MBd and 200 MBd baud rates. (top to bottom): various numbers of bits in the ADC, as listed in the header of each plot.

Fig. 8
Fig. 8

Field reconstruction algorithm with evasion of the divisive-exception.

Equations (35)

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I(t)=Re{ ρ ˜ (t) ρ ˜ * (t τ DI ) }=ρ(t)ρ(t τ DI )cos[ ρ ˜ k ρ ˜ (t τ DI ) ] Q(t)=Im{ ρ ˜ (t) ρ ˜ * (t τ DI ) }=ρ(t)ρ(t τ DI )sin[ ρ ˜ k ρ ˜ (t τ DI ) ]
q ˜ (t)=I(t)+jQ(t)= ρ ˜ (t) ρ ˜ * (t τ DI )
q ˜ k q ˜ (k τ DI )= ρ ˜ (k τ DI ) ρ ˜ * ((k1) τ DI )
q ˜ k = ρ ˜ k ρ ˜ k1 *
I k =Re ρ ˜ k ρ ˜ k1 * = ρ k ρ k1 cos( ρ ˜ k ρ ˜ k1 ) Q k =Im ρ ˜ k ρ ˜ k1 * = ρ k ρ k1 sin( ρ ˜ k ρ ˜ k1 ).
q ˜ k = I k +j Q k =Re{ ρ ˜ k ρ ˜ k1 * }+jIm{ ρ ˜ k ρ ˜ k1 * }= ρ ˜ k ρ ˜ k1 * = ρ k ρ k1 e j( ρ ˜ k ρ ˜ k1 )
q k = I k 2 + Q k 2 = ρ k ρ k1 , q ˜ k =arctan Q k I k = ρ ˜ k ρ ˜ k1
ρ ˜ ^ k = q ˜ k + ρ ˜ ^ k1 = k =0 k q ˜ k
ρ ˜ ^ k = P k exp{ j k =0 k arctan( Q k / I k ) }
ρ ^ k ρ k ρ k1 q k = ( I k 2 + Q k 2 ) 1/4
ρ ˜ ^ k = q ˜ k / ρ ˜ ^ k1 * witharbitraryinitialcondition ρ ˜ ^ 0
ρ ˜ ^ 1 = q ˜ 1 ρ ˜ ^ 0 * ; ρ ˜ ^ 2 = q ˜ 2 ρ ˜ ^ 1 * ; ρ ˜ ^ 3 = q ˜ 3 ρ ˜ ^ 2 * ..... ρ ˜ ^ k = q ˜ k ρ ˜ ^ k1 * ....
k=1,2,3,4,...:k=1: ρ ˜ ^ 1 = q ˜ 1 ρ ˜ ^ 0 * = ρ ˜ 1 ρ ˜ 0 * g ˜ 0 * ρ ˜ 0 * = ρ ˜ 1 / g ˜ 0 *
k=2: ρ ˜ ^ 2 = q ˜ 2 ρ ˜ ^ 1 * = ρ ˜ 2 ρ ˜ 1 * ( ρ ˜ 1 / g ˜ 0 * ) * = ρ ˜ 2 ρ ˜ 1 * ρ ˜ 1 * / g ˜ 0 = ρ ˜ 2 g ˜ 0
k=3: ρ ˜ ^ 3 = q ˜ 3 ρ ˜ ^ 2 * = ρ ˜ 3 ρ ˜ 2 * ( ρ ˜ 2 g ˜ 0 ) * = ρ ˜ 3 ρ ˜ 2 * ρ ˜ 2 * g ˜ 0 * = ρ ˜ 3 / g ˜ 0 *
k=4: ρ ˜ ^ 3 = q ˜ 3 ρ ˜ ^ 2 * = ρ ˜ 3 ρ ˜ 2 * ( ρ ˜ 2 g ˜ 0 ) * = ρ ˜ 3 ρ ˜ 2 * ρ ˜ 2 * g ˜ 0 * = ρ ˜ 3 / g ˜ 0 * ρ ˜ ^ 4 = q ˜ 4 ρ ˜ ^ 3 * = ρ ˜ 4 ρ ˜ 3 * ( ρ ˜ 3 / g ˜ 0 * ) * = ρ ˜ 4 ρ ˜ 3 * ρ ˜ 3 * / g ˜ 0 = ρ ˜ 4 g ˜ 0
{ ρ ˜ ^ 0 , ρ ˜ ^ 2 , ρ ˜ ^ 4 ,..., ρ ˜ ^ 2 k ,...}= g ˜ 0 { ρ ˜ 0 , ρ ˜ 2 , ρ ˜ 4 ,..., ρ ˜ 2 k ,...} { ρ ˜ ^ 1 , ρ ˜ ^ 3 , ρ ˜ ^ 5 ,..., ρ ˜ ^ 2 k +1 ,...}= g ˜ 1 { ρ ˜ ^ 1 , ρ ˜ ^ 3 , ρ ˜ ^ 5 ,..., ρ ˜ ^ 2 k +1 ,...}where g ˜ 1 1/ g ˜ 0 *
γ 0 g ˜ 0 : ρ ˜ ^ k = ρ ˜ k + γ 0 ,k=0,1,2,3,4,....
ρ ^ k = q k / ρ ^ k1 ; ρ ˜ ^ k = q ˜ k + ρ ˜ ^ k1
q ˜ k = q ˜ k o + n ˜ k q = q ˜ k o ( 1+ n ˜ k q / q ˜ k o )= q ˜ k o ( 1+ η ˜ k q ) ρ ˜ ^ k = ρ ˜ ^ k o + n ˜ k ρ = ρ ˜ ^ k o + n ˜ k ρ = ρ ˜ ^ k o ( 1+ n ˜ k ρ / ρ ˜ ^ k o )= ρ ˜ ^ k o ( 1+ η ˜ k ρ )
η k q n ˜ k q / q ˜ k o =( q ˜ k q ˜ k o )/ q ˜ k o ; η k ρ n ˜ k ρ / ρ ˜ k o =( ρ ˜ k ρ ˜ k o )/ ρ ˜ k o
ρ ^ k | ρ ˜ ^ k |= ρ ^ k o | 1+ η ˜ k ρ |= ρ ^ k o | 1+ η k ρRe +j η k ρIm | ρ ^ k o ( 1+ η k ρRe )
ρ ˜ ^ k ρ ˜ k ρ ˜ k1 * = q ˜ k ρ ˜ ^ k1 * = q ˜ k o ( 1+ η ˜ k q ) ρ ˜ ^ k1 o* ( 1+ η ˜ k1 ρ* ) = ρ ˜ ^ k o ( 1+ η ˜ k q η ˜ k1 ρ* )
η ˜ k ρ = η ˜ k q η ˜ k1 ρ*
η k ρRe = η k qRe η k1 ρRe ; η k ρIm = η k qIm + η k1 ρIm
Z{ η k ρRe }=Z{ η k qRe } z 1 Z{ η k ρRe } Z{ η k ρRe } Z{ η k qRe } = 1 1+ z 1 = z z+1
Z{ η k ρIm }=Z{ η k qIm }+ z 1 Z{ η k ρIm } Z{ η k ρIm } Z{ η k qIm } = 1 1 z 1 = z z1
ρ ˜ ^ k = ρ ˜ ^ k o ( 1+ η k ρ )= ρ ˜ ^ k o ( 1+ η k qRe +j η k qIm ) ρ ˜ ^ k o ( 1+ η k qRe ) e j η k qIm
η k ρIm = k =0 k η k qIm
H Q ( e jω )= z/(z1) | z= e jω = j 2 e jω/2 /sin( ω/2 )
η k ρRe = k =0 k (1) k k η k qRe = (1) k k =0 k (1) k η k qRe
H I ( e jω )= (1) k k =0 k (1) k e jω = z/(z+1) | z= e jω = 1 2 e jω/2 /cos( ω/2 )
ρ ˜ ^ k = ρ ˜ ^ k o ( 1+ η k qRe +j η k qIm ) ρ ˜ ^ k o ( 1+ (1) k k =0 k (1) k η k qRe )exp{ j k =0 k η k qIm }
A ˜ k = s ˜ k A ˜ k1 /| A ˜ k1 |
ε k ρ | ρ ˜ ^ k || ρ ˜ k |; ε k ϕ ρ ˜ ^ k ρ ˜ k

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