TECHNICAL FIELD

The present invention relates to an equalizer and a corresponding equalization method for a four-level pulse amplitude modulation (PAM-4) signal. In particular, the equalizer of the present invention is a nonlinear equalizer based on the Volterra model.

BACKGROUND

Next-generation ultra-high-speed short-reach optical fiber links will utilize small, cheap, and low-power consumption transceivers. These requirements for transceiver size, price and power- consumption are mainly imposed by the limited space for equipment in data centers. The transceivers should further support intra- and inter-data center connections ranging from a few hundred meters up to several tens of kilometers, respectively.

Additionally, it is preferably envisaged that 100 Gbit/s per wavelength are transmitted. This is very challenging, particularly when device size and costs are a major concern. In this case, coherent approaches are not feasible, as they require high-power consumption and expensive devices. Therefore, intensity modulation (IM) and direct detection (DD) schemes are preferred. Also the mature on-off keying modulation scheme, which is widely used in non-coherent systems, has been investigated for applications at 100 Gbit/s per wavelength speed. However, this scheme would require expensive high-bandwidth optics and electronics.

To overcome this drawback, advanced modulation schemes supported by digital signal processing (DSP) have been investigated as an alternative technology to support the envisaged 100 Gbit/s. One of the most promising of these modulation schemes is PAM-4 combined with IM and DD. However, this scheme suffers from the potential of being significantly degraded by linear and nonlinear inter-symbol interference.

A conventional PAM-4 transmission system is shown in Fig. 11. Data encoded by a forward error correction (FEC) block can be partially equalized by DSP at the transmitter side, and can be converted into an analog signal by a digital-to-analog converter (DAC). In order to decrease complexity, the data can then be equalized in the analog domain, for instance, by a continuous- time linear equalizer (CTLE).

Afterwards, this signal is amplified using a modulator driver (MD). In order to modulate the signal, a distributed feedback laser (DFB) together with an electro-absorption modulator (EAM) integrated in transmit optical subassemblies (TOSA) are often used in cheap systems. Other cheap systems include a direct-modulated laser (DML) or a vertical-cavity surface- emitting laser (VCSEL). After its modulation, the optical signal can be transmitted over different fiber types, which are selected based on requirements like distance, bit rate etc.

At the receiver side, a photo diode (PIN-positive-intrinsic negative or APD-avalanche photo diode) is typically used to detect the optical signal. The output of the photodiode is proportional to the power of the optical signal. The photodiode output is usually amplified by a trans- impedance amplifier (TIA). The photodiode and the TIA can be integrated in receiver optical subassemblies (ROSA) that may also include an automatic gain control circuit (AGC), in order to adjust the electrical signal to an analog-to-digital (ADC) input, when electronic equalization is used.

An equalizer of the receiver is used to recover signals suffering from noise and inter-symbol interference (ISI). However, before the equalizer is used, the local oscillator must be locked to the input signal, i.e. to the transmitter oscillator responsible for data clocking. The two oscillators must be synchronized. Small deviations are allowed, since it is impossible to perfectly track the transmitter clock source. Clock extraction is supported by a timing recovery (TR) block that controls ADC sampling frequency and phase. The performance of this block is strongly influenced by noise that is partly filtered out by specific filters. However, some imperfections such as bandwidth limitation and chromatic dispersion may result in a very weak timing function. Therefore, the signal used for timing recovery has to be partially compensated to enable correct ADC clocking.

The equalized signal can be used for clock extraction, in order to decrease the clock jitter. This signal can be further processed by a maximum-likelihood sequence estimator (MLSE) to improve a bit error rate (BER) before the error correction block (FEC decoder) that delivers the final decisions. The MLSE introduces an additional complexity that can be avoided in low- power consumption applications. As examples for the equalizer, Feed-forward (FFE) and decision feedback equalizers (DFE) can be found in many practical systems, while nonlinear equalizers (NLE) are less deployed, although they can bring a significant gain in some special applications.

A conventional hybrid FFE-DFE architecture is shown in Fig. 12. Here, after the ADC, the quantized data is low-pass filtered and then equalized by the FFE and DFE. Filter coefficients of both equalizers are controlled by the same error signal e. The FFE can use an oversampled signal, in order to improve the performance, and to be less sensitive to the sampling phase. Two or three samples per symbol period may improve the FFE performance, which also depends on the channel impairments. However, a performance and complexity trade-off must be done at high baud rates. The DFE is symbol-based and uses only one sample per symbol. Such one sample per symbol equalizers are often used, due to their complexity and not significant gain by introducing more samples.

The filter coefficients can be updated in blind or training mode. Normally, long training sequences are not available at the receiver side, whereas some short training sequences can be periodically inserted (better to avoid) to help the adaptive filters to converge faster and to avoid ill convergence. The ill convergence includes suboptimum convergence (which leads to suboptimum BER), false convergence (stable coefficients, but BER almost 0.5), and evolving coefficients to very high values (can go to infinity).

Instead of using training symbols, the equalizer may rely on decisions that are not error-free, but still good enough to enable the correct filter tap values setting (DD mode). The peak distortion criterion and the mean square error (MSE) criterion, also known as least mean square (LMS) algorithm, are the two most considered criteria for adaptive systems. The peak distortion one is also known as the zero forcing algorithm (ZFA) that inverts the channel transfer function, whereas the MSE criterions adds the noise spectral density before inversion. In theory, the MSE provides better performance, because the ZFA enhances noise at spectral nulls, but in adaptive systems and when noise is not significant they behave similarly.

Notably, as the present invention provides an equalizer and method especially for PAM-4 equalization, and deals with real signals, it is avoided throughout the description at hand to use conjugate complex operations in all used equations. An output signal y of the FFE equalizer in Fig. 12 can be written as: where / (with i = 0, 1, 2, N-l) is a function of the input sample vector, Wi is the i ^th filter coefficient, and the filter coefficient Wdc is responsible for a DC component. In its simplest form, the equalizer of Fig. 12 can be a linear transversal filter with outputs

A coefficient adaptation for DD-LMS algorithm is defined by where μ is called the gradient factor and denotes a small constant that controls the magnitude of the weight adjustment, <i is a decision, and y is again the FFE output. In parallel ASIC implementations, when several equalizers are executed at once, the error signal is obtained by error averaging, which improves equalizer acquisition performance, and reduces the noise in the estimate of the gradient factor. Such systems become less prone to ill convergence. The equalizer tracking speed is directly proportional to the μ value, whereas the 3-db equalizer bandwidth can be approximated by represents the symbol rate.

Especially, for binary signals, there are LMS adaptation algorithms (sign-error, sign-data, and sign-sign) that are simpler to implement, at the expense of a lower speed of convergence.

Vito Volterra, in his work dating from 1887, introduced the theory of analytic functions that was modernized and further developed by Norbert Wiener. Equalizers based on the Volterra model, i.e. equalizers using a Volterra filter, can be efficiently used for modeling, for instance, distortion in semiconductor laser diodes, the transfer function of a single mode fiber, or the nonlinear propagation inside multimode interference couplers. Such equalizers are also proposed for compensating nonlinear effects in direct-detection systems, as well as in coherent systems.

A P ^th order discrete Volterra filter with filter input x, filter output y, and memory length M can be described as wherein w _r are r order Volterra kernels. Volterra kernels are symmetric, which is exploited in this equation by considering only coefficients with non-decreasing indices fa, i.e. fa > fa-\. In this case, the number of Volterra coefficients can be calculated by

As there are coefficients in the polynomial expression, even for moderately large values of M and P the implementation complexity becomes extremely high. Thus, only lower-order Volterra filters are considered for real system equalizers. For instance, in direct-detection optical systems, a Volterra equalizer based on the restricted 3 ^rd order Volterra filter seems to be sufficient to compensate electrical and optical linear and nonlinear impairments, and is described by:

In the following the 3 order Volterra filter with M=3 (19 taps) is considered. The input vector x is defined by:

Assuming that the input signal x is a zero-mean random PAM-4 signal with unit variance, it follows that:

The Volterra filter can be updated by using DD-LMS. However, the autocorrelation matrix (ACM) Rxx, which consists of elements of the autocorrelation function (ACF), of the Volterra filter is not fully diagonalized (as shown in Fig. 13 for M=3), and the eigenvalues spread is large, the filter convergence can be very slow with multiple local minima. The full LMS algorithm can be described by

and is much more complex than DD-LMS for linear equalizers. To simplify the updating procedure, the kernels of each order can be updated at different speeds, and for the 3 ^rd order filter the following gradient vector μ can be used:

However, there is no conventional procedure, how to set the values of this gradient vector.

The fact that the Volterra filter R _xx is not fully diagonalized, can cause suboptimum convergence and in some cases acquisition failure.

For a LMS Volterra equalizer based on the nonlinear discrete Wiener model, it is proposed to use polynomial processing on the filter input vector x. In this case, x is an input vector of the polynomial processing with unit variance, and is defined by

The main problem here, however, is to find polynomials that result in an optimum acquisition performance, and especially to find coefficients of the polynomials, so as to achieve a diagonal ACM. The orthogonalization of a Volterra equalizer for PAM-4 signals has not been discussed in the existing literature.

SUMMARY In view of the above-mentioned problems, the present invention aims to improve the conventional equalizers. In particular, the present invention has the object to provide an equalizer for a PAM-4 signal with improved acquisition performance, especially in a noisy environment. That is, faster channels acquisition with at the same time decreased probability of ill acquisitions is desired. The ACM of the equalizer, when used for the PAM-4 signal, should be diagonal. Moreover, the equalizer should be as close as possible to its optimum point. Sub- optimum equalizer states should accordingly be avoided. Additionally, the equalizer should not be more complex compared than conventional equalizers.

The object of the present invention is achieved by the solution provided in the enclosed independent claims. Advantageous implementations of the present invention are further defined in the dependent claims.

A first aspect of the present invention provides an equalizer for a PAM-4 signal, the equalizer comprising a filter, a processing unit configured to apply polynomial processing to an input signal x of the equalizer, in order to output a processed signal q to the filter, an estimation unit configured to estimate a noise standard deviation σ of noise on an output signal y of the filter, and a calculation unit configured to calculate polynomial coefficients based on the estimated noise standard deviation σ, wherein the polynomial processing in the processing unit bases on orthogonal polynomials and on the calculated polynomial coefficients.

The equalizer of the first aspect is able to derive orthogonal polynomials, especially the coefficients for the orthogonal polynomials, especially for PAM-4 signals in a noisy environment. This is due to the fact that the coefficients are calculated based on the noise on the equalized signal. Specifically, the estimated noise standard deviation estimation enables the equalizer to provide the best tracking performance in time-variant channels. Further, the estimation prohibits incorrect equalizer acquisition in hard channels. With the equalizer of the first aspect, faster channel acquisition is possible, while the orthogonal polynomials decrease the probability of ill acquisitions. Accordingly, fast channel tracking is enabled (power variations, crosstalk etc.), and suboptimum states are mostly avoided.

In an implementation form of the first aspect, the filter is a Volterra filter.

The equalizer thus bases on the Volterra model. The Volterra equalizer of this implementation form can be easily provided by transforming a conventional Volterra equalizer without adding complexity.

In a further implementation form of the first aspect, the estimation unit is configured to estimate the noise standard deviation σ according to the formula wherein is the noise standard deviation of an i PAM-4 level Li defined by the formula

with i = 0, 1, 2, 3.

With the estimation unit operating in this manner, a 3 order equalizer with an optimized acquisition performance, and particularly a diagonal ACM, is obtained.

In a further implementation form of the first aspect, the estimation unit is configured to estimate the noise standard deviation PAM-4 level Li by finding a best Gaussian fitting of a histogram of an obtained data set Su

The noise standard deviation can in this manner be estimated very accurately and efficiently.

In a further implementation form of the first aspect, the estimation unit is configured to obtain the data set Si within borders Bi and Bi+i defined by the formula

for i = 0, 1, 2, 3. In a further implementation form of the first aspect, the estimation unit is configured to use mean square error for finding the best Gaussian fitting.

In a further implementation form of the first aspect, the processing unit is configured to set the noise standard deviation σ to a start value σο at the beginning of the polynomial processing, wherein

With the above value, very good acquisition results are already achieved with the equalizer from the very beginning of its operation on a PAM-4 signal. The estimation then optimizes this performance. In a further implementation form of the first aspect, a highest coefficient of any orthogonal polynomial is equal to 1.

In a further implementation form of the first aspect, a mean value of any polynomial is equal to 0, and all polynomials are orthogonal.

The above-defined rules for selecting orthogonal polynomials for the processing unit of the equalizer, lead to the best acquisition results.

In a further implementation form of the first aspect, the calculation unit is configured to calculate the polynomial coefficients according to the formulas

for i≠ j, and

for i≠ 0, wherein

for k = 0, 1, ..., n.

In a further implementation form of the first aspect, the orthogonal polynomials comprise the following polynomials

wherein a, b, c and d are polynomial coefficients calculated by the coefficient calculation unit.

These polynomial lead to an improved acquisition performance for an equalizer up to the 4 ^th order.

In a further implementation form of the first aspect, the calculating unit is configured to calculate the coefficients a, b, c and d according to

with

With the above coefficients, the best performance is achieved, particularly for an equalizer up to the 4 ^th order. In a further implementation form of the first aspect, for a third-order equalizer, the processing unit is configured to apply the polynomial processing to an input signal x of the equalizer defined according to the formula

in order to output a processed signal q defined according to the following formula

wherein a and b are polynomial coefficients calculated by the coefficient calculation unit.

A second aspect of the present invention provides a method for equalizing a PAM-4 signal, wherein the method comprises applying polynomial processing to an input signal x, in order to output a processed signal q to a filter, estimating a noise standard deviation σ of noise on an output signal y of the filter, and calculating polynomial coefficients based on the estimated noise standard deviation σ, wherein the polynomial processing bases on orthogonal polynomials and on the calculated polynomial coefficients.

The method can particularly be carried out by the equalizer of the first aspect. That is, the processing step may be carried out by the processing unit of the equalizer, the estimating step by the estimation unit of the equalizer, and the calculating step by the calculating unit of the equalizer. Implementation forms of the method according to the second aspect may add further steps according to the above-described implementation forms of the equalizer according to the first aspect, respectively. The method of the second aspect thus achieves all the advantages of the equalizer, and its implementation forms, of the first aspect. A third aspect of the present invention provides a computer program product comprising a program code for performing, when running on a computer, the method according to the second aspect or possible implementation forms thereof. Accordingly, all advantages of the method of the second aspect and its possible implementation forms are achieved.

It has to be noted that all devices, elements, units and means described in the present application could be implemented in the software or hardware elements or any kind of combination thereof. All steps which are performed by the various entities described in the present application as well as the functionalities described to be performed by the various entities are intended to mean that the respective entity is adapted to or configured to perform the respective steps and functionalities. Even if, in the following description of specific embodiments, a specific functionality or step to be performed by external entities is not reflected in the description of a specific detailed element of that entity which performs that specific step or functionality, it should be clear for a skilled person that these methods and functionalities can be implemented in respective software or hardware elements, or any kind of combination thereof.

BRIEF DESCRIPTION OF DRAWINGS

The above described aspects and implementation forms of the present invention will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which Fig. 1 shows an equalizer according to an embodiment of the present invention.

Fig. 2 shows a method according to an embodiment of the present invention.

Fig. 3 shows an equalizer according to an embodiment of the present invention.

Fig. 4 shows an ACM of an equalizer according to an embodiment of the present invention. Fig. 5 shows a diagonal vector for σ = 0 and σ =

Fig. 6 shows and estimation of σ based on histograms. Fig. 7 shows a PAM-4 experiment with an equalizer according to an embodiment of the present invention. Fig. 8 shows (a) an averaged eye diagram and (b) a performance of an equalizer according to an embodiment of the present invention. Fig. 9 shows (a) a 3 ^rd order coefficients evolution for a conventional Volterra equalizer and (b) a 3 ^rd order coefficients evolution for an equalizer according to an embodiment of the present invention. Fig. 10 shows (a) a 1 ^st order coefficients evolution for a conventional Volterra filter and

(b) a 1 ^st order coefficients evolution for an equalizer according to an embodiment of the present invention. Fig. 11 shows a PAM-4 transmission system. Fig. 12 shows a conventional equalizer. Fig. 13 shows an ACM of a conventional equalizer based on the Volterra model.

DETAILED DESCRIPTION OF EMBODIMENTS

Fig. 1 shows an equalizer 100 according to an embodiment of the present invention. The equalizer 100 is configured to equalize PAM-4 signals, especially in a noisy environment. The equalizer 100 comprises a filter 101, a processing unit 102, an estimation unit 103, and a calculation unit 104. The filter 101 bases preferably on the Volterra model or Wiener model, i.e. is preferably a Volterra filter (or Wiener filter). The filter 101 is arranged to receive a signal q from the processing unit 102, and to output an equalized signal y.

The processing unit 102 is configured to apply polynomial processing to an input signal x of the equalizer 100. After applying the polynomial processing, the processing unit 102 outputs the processed signal q to the filter 101. The polynomial processing in the processing unit 102 bases particularly on orthogonal polynomials, and on calculated polynomial coefficients. These coefficients are calculated, and provided to the processing unit 102, by the calculation unit 104. To this end, the estimation unit 103 is configured to estimate a noise standard deviation σ of noise on the output signal y of the filter 101, and to provide this estimation to the calculation unit 104. The calculation unit 104 is configured to calculate the polynomial coefficients based on the provided estimated noise standard deviation σ. Fig. 2 shows a corresponding method 200 according to an embodiment of the present invention for equalizing a PAM-4 signal. The method comprises a step 201, in which polynomial processing is applied to an input signal x, in order to output a processed signal q to a filter 101. This step 201 may be carried out by the processing unit 102 of equalizer 100. The method comprises a further step 202, in which a noise standard deviation σ of noise on an output signal y of the filter is estimated. This step 202 may be carried out by the estimating unit 103. The method comprises a further step 203, in which polynomial coefficients are calculated based on the estimated noise standard deviation σ. This step 203 may be carried out by the calculation unit 104. The polynomial processing carried out in step 201 bases on orthogonal polynomials, and on the polynomial coefficients calculated in step 203 (the feedback of these coefficients is indicated in Fig. 2 by the arrow from 203 to 201).

Fig. 3 shows another embodiment of an equalizer 100 according to the present invention, which builds on the embodiment of Fig. 1. The equalizer 100 includes further a decision unit 301, a LMS unit 302, and a subtractor unit 303.

The decision unit 301 is configured to receive the filter output signal y, and to output a decision d. The subtractor unit 303 is configured to combine this decision d with the negative of the output signal y of the filter 101, in order to generate an error signal e. The LMS unit 302 is configured to receive the error signal e, and further to receive a gradient factor μ, which denotes a small constant that controls a magnitude of a filter weight adjustment, an ACM diagonal s, and the signal q that is input into the filter 101. The LMS unit 302 is further configured to output a filter coefficient w to the filter 101.

For the polynomial processing, which is applied by the processing unit 102, orthogonal polynomials up to n ^th order may be defined with coefficients according to: Odd-order polynomials have only odd coefficients, and even-order polynomials have only even coefficients. Coefficients for polynomials up to the 4 ^th order can be represented by the following matrix:

The coefficients are selected by the calculation unit 104 of equalizer 100 preferably such that the following equations hold (self-orthogonal):

Assuming for the processing unit 102 specifically a zero-mean PAM-4 input signal where is additive Gaussian noise with variance and the PAM symbols are

uncorrected, it follows:

In this case, the orthogonal polynomials can be found according to: i.e., the orthogonal polynomials used by the processing unit 102 comprise the following polynomials.

In the above equation (5), a, b, c and d are the polynomial coefficients, which are calculated by the coefficient calculation unit 104 of the equalizer 100.

The following rules may be used to calculate coefficients and derive polynomials. Firstly, the mean value of any polynomial, odd or even, is always 0. Secondly, all, i.e. odd and even, polynomials are always orthogonal. The orthogonality conditions generate more equations, which can be used to calculate the coefficients. The orthogonality is preferably checked only between odd or between even polynomials. Thirdly, the highest coefficient is always 1. Fourthly, the second polynomial mean value must be 0.

From Ε{x ² +α}=0 it then follows that:

From E{(x ³ +bx)x}=0 it further follows that

From E{x ⁴ +cx ² +d}=0 and E{(x ⁴ +cx ² +d) (x ² +a)}=0 it further follows that:

Thereby, according to the relations given above in (3), it follows that:

In the equalizer 100, the kernels defined in (1) are transformed by the polynomial processing unit 102, which operates based on the above-found polynomials and coefficients. For example, the output vector q after polynomial processing in the processing unit 102 is (for 3 ^rd order, i.e. M=3)

The diagonal autocorrelation matrix is then defined by R _qq . Finding the inverse matrix is trivial from the diagonal vector of R _qq that is

A new vector can now be defined:

Thus, equation (2) can be simplified and the updating algorithm is

In the equalizer 100, each kernel is preferably controlled by its own gradient factor μ, and the final speed is mainly influenced by the gradient factor μ. The effect of the equalizer 100 according to an embodiment of the present invention is visible in Fig. 4, representing the ACM after polynomial processing (here M=3). The μ working range can be adjusted to specific cases.

A simplified Volterra filter uses for the coefficients update, whereas the Wiener equalizer

updates the coefficients by The results of multiplication can, for example, be stored

in a look-up table so that it does not introduce noticeable complexity. Also, the number of multiplications in both cases is identical (only different number of additions), and it can be concluded that these two approaches have similar complexity O(M ³ ).

At lower BERs, the parameter σ can be freely set to zero, whereas setting σ to larger values improves stability and accelerate acquisition at higher BERs (ISI and noise scenario). The diagonal vectors for M=[35,5,7] (i.e. 35 linear coefficients, five 2 ^nd order and seven 3 ^rd order coefficients) and is shown in Fig. 5 (for PAM-4 levels

The noise variance affects the updating algorithm, and σ can be estimated and set correctly.

Using provides suboptimum, but very good performance. The equalizer 100

preferably starts with this value, and then stabilizes the coefficients by noise estimation. The acquisition at the beginning may be a bit slower than in the optimum case, when the noise standard deviation σ is better known, but after the acquisition phase the channel may vary (PMD, power variations etc.), and the filter should track such variations.

The tracking performance depends on the gradient factor μ and the standard deviation σ. The gradient factor μ is normally unchanged after the acquisition period. Nevertheless, the tracking capability can be improved, when the polynomial coefficients are calculated using the true value of σ. Therefore, the noise standard deviation σ is - as already shown in and described with respect to the Figs. 1 and 2 - estimated by the estimation unit 103 on the equalized signal y after the filter 101. The estimated value is provided to the calculation unit 104, which calculates the polynomial coefficients using the above formulas (3), (4), and (5). Later, formula (6) is used to finally orthogonalize the equalizer 100. The noise standard deviation σ is particularly estimated by using

wherein is the noise standard deviation of an i PAM-4 level Li defined by the formula

with i = 0, 1, 2, 3.

Five borders Bi and Bi+i are introduced and defined by the formula

for i = 0, 1, 2, 3.

Sets of data within the borders are denoted as Si with i = 0, 1, 2, 3. The mean value and standard deviation for each PAM-4 level can be easily estimated. The mean values are estimated accurately, while deviations are underestimated (smaller than they are). Therefore, σ is preferably increased in small steps, and the best Gaussian fitting is preferably found with histograms - as presented by the symbols in Fig. 6. The best fitting uses mean square error to detect the best σ value. That is, the estimation unit 103 is configured to use mean square error for finding the best Gaussian fitting. To get histograms, only 11 bins are enough (see cycles in Fig. 6). After pdf adjustment, σ can be obtained very accurately, and the corresponding pdfs are also shown in the same figure. With the new σ value the algorithm performs the best.

When channel conditions are heavy (noise, distortion, clock jitter etc.) increasing the number of linear taps may cause ill acquisition. The coefficients at the tails may not converge, which results in suboptimum performance or even results in a case, where the acquisition completely fails. Mostly, noise is the main reason for such a behavior. The same behavior was, however, observed in experiments, not only in noise-dominated scenarios, but also in heavy nonlinear systems with pre- and post-taps that do not decrease quasi exponentially. This problem can be easily caused, even in slightly distorted channels (low noise), when the number of taps is set to a very high value (e.g. 1000).

In order to overcome this problem the following procedure is implemented for the DD-LMS algorithm of the LMS unit 302 shown in Fig. 3. 1. Normalize the signal power to 1

2. Set PAM-4 levels to

3. Set all coefficients to 0 and the central tap to 1.

4. Update only N of M linear coefficients where 3 < N< 7 and 5xl0 ^-4 < μ < 10 ^-3 .

5. After K updates (103 < K < 104) update all coefficients with 10 ^-4 < μ < 5xl0 ^-4 . 6. If channel is static, set μ < 10 ^-4 .

The above procedure enables the correct acquisition with a high probability.

A 56 GBaud experiment was carried out with high-bandwidth components, the setup of which is shown in Fig. 7, in order to get a clear PAM-4 signal after equalization. In the experimental setup of Fig. 7, a bit pattern generator (BPG) generates binary symbols that are combined to get PAM-4 symbols. After signal amplification by a modulator driver (MD), TOSA outputs an optical PAM-4 signal, which is optically amplified before a ROSA detects the optical signal. The electrical PAM-4 signal is electrically amplified and captured by the real-time scope. As the transmitter and the real-time scope clock sources were not synchronized, the clock recovery based on Mueller and Miiller phase detector (M&MPD) was used in a second-order phase- locked loop (PLL) with a 4-MHz bandwidth to compensate for the frequency offset and timing jitter. The signal is equalized and Gray BER is estimated.

Despite high-bandwidth components used in the experiment, some nonlinearities are generated. These nonlinearities are visible in Fig. 8. Fig. 8 shows in (a) the averaged eye diagram over three symbol intervals at the input power of 0 dBm, after a linear equalizer with 55 taps and one sample per symbol.

Volterra equalizers were tested up to the 4 ^th order with parameters M=[Mi,M ₂ ,M3,M4]=[35, 7,7,7]. The equalization results are shown in Fig. 8 in (b), wherein V _n denotes an n order Volterra filter. It can be seen that the 3 order kernels bring more gain at lower BER values, whereas the 4 ^th order kernels are irrelevant for the equalization performance.

Taps evolution of Volterra and Wiener filters for M=[35,5,7], and Pin=-3 dBm (σ was set to were monitored. At the beginning, all coefficients were set to 0, and wi(17) to 1.

The tap evolution is very specific and unstable for a conventional Volterra equalizer that is visible in Fig. 9 (a) representing the 3 ^rd order kernels. Even after two million updates (symbols), the 3 ^rd order kernels are not settled well, and likely cause 1 ^st order kernels instabilities. On the other hand side, the equalizer 100 is very stable after 5x10 ⁵ symbols, as is indicated in Fig. 9 (b) showing the 3 ^rd order kernels evolution. Compared with the equalizer 100, the conventional Volterra equalizer requires approximately a four times longer time to stabilize taps. It is interesting that both equalizers after 10 ⁶ symbols have similar BER performance. At lower BERs, the parameter σ can be freely set to zero, whereas setting σ to larger values improves stability and accelerate acquisition at higher BERs (ISI and noise scenario).

The 2 ^nd order coefficients are not influenced. However, the 1 ^st order coefficients evolution is similar to the 3 ^rd order kernels. Much more acquisition time is required for the conventional Volterra equalizer shown in Fig. 10 in (a) as for the equalizer 100 shown in Fig. 10 in (b).

The present invention has been described in conjunction with various embodiments as examples as well as implementations. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed invention, from the studies of the drawings, this disclosure and the independent claims. In the claims as well as in the description the word "comprising" does not exclude other elements or steps and the indefinite article "a" or "an" does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.