EQUALIZER FOR FOUR-STAGE IMPULSE AMPLITUDE MODULATION

Abstract

Equalizer (100) for a four-stage pulse amplitude modulation signal, PAM-4 signal, wherein the equalizer (100) has: a filter (101), a processing unit (102) designed to apply polynomial processing to an input signal x of the equalizer in order to output a processed signal q to the filter (101), an estimation unit (103) designed to estimate a noise standard deviation σ of noise at an output signal y of the filter (101), and a computation unit (104) designed to calculate polynomial coefficients based on the estimated noise standard deviation σ, where the polynomial processing in the processing unit (102) is based on orthogonal polynomials and on the calculated polynomial coefficients.

Description

TECHNICAL FIELD

[0001] The present invention relates to an equalizer and a corresponding equalization method for a four-level pulse amplitude modulation (PAM-4) signal. The equalizer of the present invention is, in particular, a nonlinear equalizer based on the Volterra model. BACKGROUND

[0002] Next-generation ultra-high-speed, short-reach fiber optic connections will use small, inexpensive, and low-power transceivers. These requirements for transceiver size, price, and power consumption are primarily driven by the limited equipment space in data centers. Furthermore, the transceivers should support connections within and between data centers, ranging from a few hundred meters to tens of kilometers.

[0003] Additionally, 100 Gbit / s per wavelength is preferably envisioned. This is a significant challenge, especially when device size and cost are a primary concern. In this case, coherent approaches are not feasible due to their high power consumption and expensive equipment requirements. Therefore, intensity modulation (IM) and direct detection (DD) schemes are preferred. The mature on / off duty cycle modulation scheme, widely used in non-coherent systems, has also been investigated for applications with a speed of 100 Gbit / s per wavelength. However, this scheme would require expensive, high-bandwidth optics and electronics.

[0004] To overcome this drawback, advanced modulation schemes supported by digital signal processing (DSP) have been investigated as an alternative technology to support the planned 100 Gbps. One of the most promising of these modulation schemes is PAM-4 in combination with IM and DD. However, this scheme suffers from its potential being significantly compromised by linear and nonlinear intersymbol interference.

[0005] A conventional PAM-4 transmission system is in Fig. 11. Data encoded by a forward error correction (FEC) block can be partially equalized by DSP at the transmitter end and converted to an analog signal by a digital-to-analog converter (DAC). To reduce complexity, the data can then be equalized in the analog domain, for example, by a continuous-time linear equalizer (CTLE).

[0006] This signal is then amplified using a modulator driver (MD). To modulate the signal, low-cost systems often use a distributed feedback laser (DFB) together with an electro-absorption modulator (EAM) integrated into transmit optical subassemblies (TOSA). Other low-cost systems include a direct-modulated laser (DML) or a vertical-cavity surface-emitting laser (VCSEL). After modulation, the optical signal can be transmitted over various fiber types, selected based on requirements such as distance, bit rate, etc.

[0007] On the receiver side, a photodiode (a positive-intrinsic negative PIN or an APD avalanche photodiode) is typically used to detect the optical signal. The photodiode output is proportional to the optical signal power. The photodiode output is typically amplified by a transimpedance amplifier (TIA). The photodiode and TIA may be integrated into receiver optical subassemblies (ROSA), which may also include an automatic gain control circuit (AGC) to adapt the electrical signal to an analog-to-digital (ADC) input when using electronic equalization.

[0008] A receiver equalizer is used to restore signals suffering from noise and inter-symbol interference (ISI). Before using the equalizer, however, the local oscillator—the transmitter oscillator responsible for data clocking—must be applied to the input signal. The two oscillators must be synchronized. Small deviations are acceptable, as it is impossible to perfectly track the transmitter clock source. Clock extraction is supported by a Timing Recovery (TR) block, which controls the ADC sampling frequency and phase. The performance of this block is strongly influenced by noise, which is partially filtered out by specific filters. However, some imperfections, such as bandwidth limitation and chromatic dispersion, can lead to very poor timing performance.Therefore, the signal used for timing recovery must be partially compensated to enable correct ADC clocking.

[0009] The equalized signal can be used for clock extraction to reduce clock jitter. This signal can be further processed by a maximum likelihood sequence estimator (MLSE) to improve the bit error rate (BER) before the error correction block decoder (FEC decoder), which provides the final decisions. The MLSE introduces additional complexity that can be avoided in low-power applications.

[0010] Examples of equalizers include feed-forward equalizers (FFE) and decision-feedback equalizers (DFE), which are found in many practical systems, while non-linear equalizers (NLE) are less commonly used, although they can provide significant benefits in some specific applications.

[0011] A conventional hybrid FFE-DFE architecture is in Fig. 12. Here, the quantized data is low-pass filtered after the ADC 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 to improve performance and be less sensitive to the sampling phase. Two or three samples per symbol period can improve FFE performance, which also depends on the channel impairments. However, at high baud rates, a trade-off must be made between performance and complexity. The DFE is symbol-based and uses only one sample per symbol. Such equalizers with one sample per symbol are widely used due to their complexity and the fact that they do not provide significant gain by introducing additional samples.

[0012] The filter coefficients can be updated in blind or training mode. Typically, long training sequences are not available on the receiver side, whereas a few short training sequences can be inserted periodically (which is best avoided) to help the adaptive filters converge faster and avoid poor convergence. Poor convergence includes suboptimal convergence (leading to a suboptimal BER), false convergence (stable coefficients but a BER close to 0.5), and coefficients that escalate to very high values ​​(which can go to infinity).

[0013] Instead of using training symbols, the equalizer can rely on decisions that are not error-free but still good enough to allow the correct setting of filter tap values ​​(DD mode). The peak distortion criterion and the mean square error (MSE) criterion, also known as the least mean square (LMS) algorithm, are the two most commonly considered criteria for adaptive systems. Peak distortion is also known as the zero-forcing (ZFA) algorithm, which inverts the channel transfer function, while the MSE criterion adds the noise spectral density before the inversion. In theory, MSE offers better performance because the ZFA increases the noise at spectral nulls, but in adaptive systems and for insignificant noise, they behave similarly.

[0014] Since the present invention provides an equalizer and method specifically for PAM-4 equalization and deals with real signals, it should be noted that the present description consistently avoids using conjugate complex operations in all equations used.

[0015] An output signal y of the FFE equalizer in Fig. 12 can be written as follows: y(n)=wdc+∑i=0N−1wifi(x(n)) where f i (with i = 0, 1, 2, ..., N-1) is a function of the input sample vector, w i is the i-th filter coefficient and the filter coefficient w dc is responsible for a DC component. In its simplest form, the equalizer can be Fig. 12 be a linear transversal filter with the following outputs: y(n)=wdc+∑i=−KKwixn−i

[0016] A coefficient adjustment for the DD-LMS algorithm is defined by wj(k+1)=wj(k)+μekfj(x(k)) wdc(k+1)=wdc(k)+μek,ek=dk−yk, where µ is called the gradient factor and is a small constant that controls the size of the weight adjustment, d represents a decision, and y is again the FFE output.

[0017] In parallel ASIC implementations, when multiple equalizers are running simultaneously, the error signal is obtained by error averaging, which improves the detection performance of the equalizer and reduces noise in the gradient factor estimation. Such systems are less prone to poor convergence. The tracking speed of the equalizer is directly proportional to the µ value, whereas the 3 dB equalizer bandwidth is determined by µf. s can be approximated, where f srepresents the symbol rate. Especially for binary signals, there are LMS (character-error, character-data, and character-character) adaptation algorithms that are easier to implement at the expense of a lower convergence speed.

[0018] In his 1887 work, Vito Volterra introduced the theory of analytic functions, which was modernized and further developed by Norbert Wiener. Equalizers based on the Volterra model, i.e., equalizers with a Volterra filter, can be used efficiently to model, for example, distortion in semiconductor laser diodes, the transfer function of a single-mode fiber, or nonlinear propagation in multimode interference couplers. Such equalizers are also proposed for compensating nonlinear effects in direct-detection systems as well as in coherent systems.

[0019] A discrete Volterra filter of P-th order with a filter input x, a filter output y and a memory length M can be described as y(k)=wdc+∑r=1P∑k1=0M−1…∑kr−kr−1M−1wr(k1,k2,…,kr)x(k−k1)⋯x(k−kr) where w r are r-th order Volterra kernels. Volterra kernels are symmetric, which is exploited in this equation by only considering coefficients with non-decreasing indices k r , ie k r ≥ k r-1 , can be considered. In this case, the number of Volterra coefficients can be calculated by 1+∑r=1P(M+r−1r)

[0020] Since in the polynomial expression O(M P) coefficients are present, the implementation complexity becomes extremely high even for moderately large values ​​of M and P. Consequently, only lower-order Volterra filters are considered for equalizers of a real system. For example, in direct-detection optical systems, a Volterra equalizer based on a limited third-order Volterra filter appears to be sufficient to compensate for electrical and optical linear and nonlinear impairments and is described by: y(k)=wdc+∑k1=0M1−1w1(k1)x(k−k1)+∑k1=0M2−1∑k2=k1M2−1w2(k1,k2)x(k−k1)x (k−k2)+∑k1=0M3−1∑k2=k1M3−1∑k3=k2M3−1w3(k1,k2,k3)x(k−k1)x(k−k2)x(k−k3)

[0021] The following considers the third-order Volterra filter with M=3 (19 taps). The input signal vector x is defined by: x(k)=[x(k),x(k−1),x(k−2),x2(k),x(k)x(k−1),x(k)x(k−2),x2(k−1),x(k−1)x (k−2),x2(k−2),x3(k),x2(k)x(k−1),x2(k)x(k−2),…,x2(k−1)x(k−2),x3(k−2)]T

[0022] Assuming that the input signal x is a mean-free random PAM-4 signal with unit variance, this gives: E{xn}=0.5(an+bn),a=1 / (5),b=3 / (5)E{xn}=0, for oddE{x2}=1,E{x4}=4125,E{x6}=7325

[0023] The Volterra filter can be updated using DD-LMS. However, the autocorrelation matrix (ACM) R xx of the Volterra filter, which consists of elements of the autocorrelation function (ACF), is not fully diagonalized (as in Fig. 13 for M=3), the distribution of eigenvalues ​​is large, and filter convergence can be very slow for multiple local minima. The full LMS algorithm can be described by w(k+1)=w(k)+μek((Rxx))−1x(k)wdc(k+1)=wdc(k)+μek and is much more complex for linear equalizers than DD-LMS. To simplify the update procedure, the kernels of each order can be updated at different rates, and the following gradient vector µ can be used for the third-order filter: μ=[μdc,μ1,μ2,μ3]T

[0024] However, there is no conventional procedure for setting the values ​​of this gradient vector.

[0025] The fact that the Volterra filter R xx is not fully diagonalized can cause suboptimal convergence and, in some cases, acquisition failure.

[0026] For an 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 x(k)=[x(k),x(k−1),x(k−2),…,x(k−M+1)]T

[0027] However, the main problem here is to determine polynomials that lead to optimal detection performance and, in particular, to determine coefficients of the polynomials to achieve diagonal ACM.

[0028] The orthogonalization of a Volterra equalizer for PAM-4 signals has not yet been discussed in the existing literature.

[0029] Document EP 1 755 296 A2 discloses a method for identifying and modeling nonlinearities in communication channels, particularly optical communication channels. A channel is generally modeled as a summation of linear and nonlinear terms with memory. The terms are functions of the input to the channel with respect to time, such as a sequence of input bits to the channel. In one embodiment, the last 5 input bits to the optical channel are used to access a value in a lookup table. The accessed value is then compared with an actual value received from the channel. The difference between the value in the table and the actual channel value can be used to correct the value in the table, for example, with an LMS (Least Means Squared) algorithm.When the lookup table and channel converge, the lookup table contains a model of the channel with memory, which can model nonlinearities. A nonlinear channel with memory can also be modeled in terms of Volterra kernels, which can be converted into a lookup table model using the Hadamard transform. The Volterra kernel representation also has the added advantage of being able to represent a lookup table with N table entries in at most N-1 Volterra kernels. In many cases, only a few Volterra kernels are required to model the channel's behavior. For example, embodiments are disclosed in which two kernels can model a nonlinear fiber optic channel operating at a wavelength of 850 nm over a multimode channel with a data rate of 1 gigabit per second.The nonlinear channel estimator can be used in a feedback equalizer or in combination with a Viterli decoder. SUMMARY

[0030] In view of the above-mentioned problems, the present invention aims to improve conventional equalizers. A specific object of the present invention is to provide an equalizer for a PAM-4 signal with improved detection performance, especially in a noisy environment. This means that faster channel detection while reducing the likelihood of poor detections is desired. The ACM of the equalizer should be diagonal when used for the PAM-4 signal. Furthermore, the equalizer should be as close as possible to its optimal point. Suboptimal equalizer states should therefore be avoided. Furthermore, the equalizer should not be more complex than conventional equalizers.

[0031] The object of the present invention is achieved by the solution provided in the appended independent claims. Advantageous implementations of the present invention are further defined in the dependent claims.

[0032] 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 to output a processed signal q to the filter, an estimation unit configured to estimate a noise standard deviation σ of the noise at an output signal y of the filter, and a calculation unit configured to calculate polynomial coefficients based on the estimated noise standard deviation σ, the polynomial processing in the processing unit being based on orthogonal polynomials and on the calculated polynomial coefficients.

[0033] The equalizer of the first aspect is capable of deriving orthogonal polynomials, particularly the coefficients for the orthogonal polynomials, especially for PAM-4 signals in a noisy environment. This is because the coefficients are calculated based on the noise of the equalized signal. In particular, the estimated estimation of the noise standard deviation enables the equalizer to achieve the best tracking performance in time-varying channels. Furthermore, the estimation prevents false equalizer detection in hard channels. With the equalizer of the first aspect, faster channel detection is possible, while the orthogonal polynomials reduce the probability of poor detections. Accordingly, fast channel tracking (power fluctuations, crosstalk, etc.) is possible, and suboptimal conditions are largely avoided.

[0034] In one implementation of the first aspect, the filter is a Volterra filter.

[0035] The equalizer is thus based on the Volterra model. The Volterra equalizer in this implementation can be easily implemented by converting a conventional Volterra equalizer without introducing any complexity.

[0036] In a further implementation of the first aspect, the estimation unit is designed to calculate the noise standard deviation σ according to the formula σ=14∑i=03σi to estimate, where σ i the noise standard deviation of an i-th PAM-4 level L i which is given by the formula Li=−35+2i5 is defined, with i = 0, 1, 2, 3.

[0037] With an estimator operating in this way, a third-order equalizer with optimized detection performance and, in particular, a diagonal ACM is obtained.

[0038] In a further implementation of the first aspect, the estimation unit is designed to calculate the noise standard deviation σ i the i-th PAM-4 level L i by finding the best Gaussian fit of a histogram of a obtained data set S i is determined.

[0039] The noise standard deviation can be estimated very accurately and efficiently in this way.

[0040] In a further implementation of the first aspect, the estimation unit is designed to process the data set S i within limits B i and B i+1 which is given by the formula Bi=−45+2i5 are defined for i = 0, 1, 2, 3.

[0041] In a further implementation of the first aspect, the estimation unit is designed to use the mean square deviation in determining the best Gaussian fit.

[0042] In a further implementation of the first aspect, the processing unit is designed to set the noise standard deviation σ at the beginning of the polynomial processing to a starting value σ0, where σ0=125

[0043] With the above value, the equalizer achieves very good detection results right from the start of its operation on a PAM-4 signal. The estimation then optimizes this performance.

[0044] In a further implementation of the first aspect, the highest coefficient of any orthogonal polynomial is 1.

[0045] In a further implementation of the first aspect, a mean value of each polynomial is 0, and all polynomials are orthogonal.

[0046] The rules defined above for selecting orthogonal polynomials for the equalizer processing unit lead to the best detection results.

[0047] In a further implementation of the first aspect, the calculation unit is designed to calculate the polynomial coefficients according to the formulas E[pi(x)pj(x)]=0 for i ≠ j, and E[pi(x)]=0 for i ≠ 0, where pk(x)=akkxk+ak−1kxk−1+…+a1kx1+a0k for k = 0, 1, ..., n.

[0048] In a further implementation of the first aspect, the orthogonal polynomials include the following polynomials G0(x)=1 G1(x)=x G2(x)=x2+a G3(x)=x3+bx G4(x)=x4+cx2+d where a, b, c and d are polynomial coefficients calculated by the coefficient calculation unit.

[0049] These polynomials lead to improved detection performance for an equalizer up to the fourth order.

[0050] In a further implementation of the first aspect, the calculation unit is designed to calculate the coefficients a, b, c and d according to a=−Ex2 b=−Ex4Ex2 c=−Ex2Ex4−Ex6(Ex2)2−Ex4 d=(Ex4)2−Ex6Ex2(Ex2)2−Ex4 to calculate with Ex2=1+σ2 Ex4=4125+6σ2+3σ4 Ex6=7325+61525σ2+45σ4+15σ6.

[0051] The above coefficients provide the best performance, especially for an equalizer up to the fourth order.

[0052] In a further implementation of the first aspect, for a third-order equalizer, the processing unit is designed to apply polynomial processing to an input signal x of the equalizer, which is calculated according to the formula x(k)=[x(k),x(k−1),x(k−2),x2(k),x(k)x(k−1),x(k)x(k−2),x2(k−1),x(k−1)x (k−2),x2(k−2),x3(k),x2(k)x(k−1),x2(k)x(k−2),…,x2(k−1)x(k−2),x3(k−2)]T is defined to output a processed signal q which is expressed according to the following formula q(k)=[x(k),x(k−1),x(k−2),x2(k)+a,x(k)x(k−1),x(k)x(k−2),x2(k−1)+a,x(k−1)x(k−2),x2(k−2 )+a,x3(k)+bx(k),(x2(k)+a)x(k−1),(x2(k)+a)x(k−2),…,(x2(k−1)+a)x(k−2),x3(k−2)+bx(k−2)]T where a and b are polynomial coefficients calculated by the coefficient calculation unit.

[0053] A second aspect of the present invention provides a method for equalizing a PAM-4 signal, the method comprising: applying polynomial processing to an input signal x to output a processed signal q to a filter, estimating a noise standard deviation σ of the noise at an output signal y of the filter, and calculating polynomial coefficients based on the estimated noise standard deviation σ, wherein the polynomial processing is based on orthogonal polynomials and on the calculated polynomial coefficients.

[0054] The method can be performed, in particular, by the equalizer of the first aspect. This means that the processing step can be performed by the processing unit of the equalizer, the estimation step by the estimation unit of the equalizer, and the calculation step by the calculation unit of the equalizer. Implementations of the method according to the second aspect can add further steps according to the above-described implementations of the equalizer according to the first aspect. The method of the second aspect thus achieves all the advantages of the equalizer of the first aspect and its implementations.

[0055] A third aspect of the present invention provides a computer program product comprising program code for performing, when run on a computer, the method according to the second aspect or possible implementations thereof. Accordingly, all the advantages of the method of the second aspect and its possible implementations are achieved.

[0056] It should be noted that all devices, elements, units, and means described in the present application can be implemented in software or hardware elements or any combination thereof. All steps performed by the various units described in the present application, as well as the functionalities to be performed by the various units, are intended to mean that the respective unit is adapted or configured to perform the respective steps and functionalities. Even if a particular functionality or step that is to be performed is not specifically mentioned in the following description of specific embodiments,to be performed by external units is not reflected in the description of a specific executable element of that unit that performs that specific step or functionality, it should be clear to a person skilled in the art that those methods and functionalities may be implemented in corresponding software or hardware elements or any combination thereof. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] The above-described aspects and implementations of the present invention are explained in the following description of specific embodiments with reference to the accompanying drawings. 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=125. Fig. Figure 6 shows an estimate of σ based on histograms. Fig. Figure 7 shows a PAM-4 experiment with an equalizer according to an embodiment of the present invention. Fig. Figure 8 shows (a) an averaged eye diagram and (b) a performance of an equalizer according to an embodiment of the present invention. Fig. Figure 9 shows (a) an evolution of third-order coefficients for a conventional Volterra equalizer and (b) an evolution of third-order coefficients for an equalizer according to an embodiment of the present invention. Fig. Figure 10 shows (a) an evolution of first-order coefficients for a conventional Volterra filter and (b) an evolution of first-order coefficients for an equalizer according to an embodiment of the present invention. Fig. Figure 11 shows a PAM-4 transmission system. Fig. 12 shows a conventional equalizer. Fig. Figure 13 shows an ACM of a conventional equalizer based on the Volterra model. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0058] Fig. Figure 1 shows an equalizer 100 according to an embodiment of the present invention. The equalizer 100 is designed to equalize PAM-4 signals, particularly in noisy environments. The equalizer 100 comprises a filter 101, a processing unit 102, an estimation unit 103, and a calculation unit 104.

[0059] The filter 101 is preferably based on the Volterra model or the Wiener model, ie, it is preferably a Volterra filter (or a Wiener filter). The filter 101 is configured to receive a signal q from the processing unit 102 and output an equalized signal y.

[0060] Processing unit 102 is configured to apply polynomial processing to an input signal x of equalizer 100. After applying the polynomial processing, processing unit 102 outputs the processed signal q to filter 101. The polynomial processing in processing unit 102 is based in particular on orthogonal polynomials and calculated polynomial coefficients. These coefficients are calculated by calculation unit 104 and then provided by the calculation unit 104 to processing unit 102.

[0061] For this purpose, the estimation unit 103 is configured to estimate a noise standard deviation σ of the noise on the output signal y of the filter 101 and to provide this estimate 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.

[0062] 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 to output a processed signal q to a filter 101. This step 201 can be performed by the processing unit 102 of the equalizer 100. The method comprises a further step 202 in which a noise standard deviation σ of the noise at an output signal y of the filter is estimated. This step 202 can be performed by the estimation 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 can be performed by the calculation unit 104.The polynomial processing performed in step 201 is based on orthogonal polynomials and on the polynomial coefficients calculated in step 203 (the feedback of these coefficients is shown in . Fig. 2 indicated by the arrow from 203 to 201).

[0063] Fig. Figure 3 shows another embodiment of an equalizer 100 according to the present invention, which is based on the embodiment of Fig. 1. The equalizer 100 further comprises a decision unit 301, an LMS unit 302 and a subtraction unit 303.

[0064] The decision unit 301 is configured to receive the filter output signal y and output a decision d. The subtraction unit 303 is configured to combine this decision d with the negative of the output signal y of the filter 101 to generate an error signal e. The LMS unit 302 is configured to receive the error signal e and, in addition, a gradient factor µ, which denotes a small constant that controls the size of a filter weighting adjustment, an ACM diagonal s, and the signal q input to the filter 101. The LMS unit 302 is further configured to output a filter coefficient w to the filter 101.

[0065] For the polynomial processing applied by the processing unit 102, orthogonal polynomials up to the nth order with coefficients can be defined according to: pk(x)=akkxk+ak−1kxk−1+…+a1kx1+a0k,k=0,1,…,n

[0066] Odd-order polynomials have only odd coefficients, and even-order polynomials have only even coefficients. Coefficients for polynomials up to the fourth order can be represented by the following matrix: [a0000000a11000a020a22000a130a330a040a240a44]

[0067] The coefficients are preferably selected by the calculation unit 104 of the equalizer 100 so that the following equations are valid (self-orthogonal): E[pi(x)pj(x)]=0 for i ≠ j, and E[pi(x)]=0 for i ≠ 0, where pk(x)=akkxk+ak−1kxk−1+…+a1kx1+a0k for k = 0, 1, ..., n.

[0068] If we assume for the processing unit 102 a mean-free PAM-4 input signal x=s+η, where E{s 2}=1, η is an additive Gaussian noise with variance σ 2 and the PAM symbols are not correlated, then it follows that: E{s2}=Es2=1,E{s4}=Es4=4125,E{s6}=Es6=7325,E{η2}=Eη2=σ2,E{η4}=Eη4=3σ4,E{η6}=Eη6=1 5σ6,E{x2}=Ex2=1+σ2,E{x4}=Ex4=Es4+Eη4+6Es2Eη2,E{x6}=Ex6=Es6+Eη6+15Es2Eη4+15Es4Eη2,

[0069] In this case, the orthogonal polynomials can be calculated according to: [1000001000−E2201000−Ex4 / Ex2010Ex2(Ex2Ex2−Ex6) / ((Ex2)2−Ex4)−Ex40−(Ex2Ex2−Es6) / ((Ex2)2−Ex4)01] determine, ie the orthogonal polynomials used by the processing unit 102 have the following polynomials. G0(x)=1G1(x)=xG2(x)=x2+aG3(x)=x3+bxG4(x)=x4+cx2+d

[0070] In equation (5) above, a, b, c and d are the polynomial coefficients calculated by the coefficient calculation unit 104 of the equalizer 100.

[0071] The following rules can be used to calculate coefficients and differentiate polynomials. First, the mean of any polynomial, whether odd or even, is always 0. Second, all polynomials—that is, odd and even polynomials—are always orthogonal. The orthogonality conditions generate more equations that can be used to calculate coefficients. Orthogonality is preferably tested only between odd or even polynomials. Third, the highest coefficient is always 1. Fourth, the mean of the second polynomial must be 0.

[0072] From E{x 2 +a}=0 then the following results: a=−Ex2.

[0073] From E{ (x 3 +bx)x}=0 it further follows that: b=−Ex4Ex2.

[0074] From E{x 4 +cx 2 +d}=0 and E{ (x 4+cx 2 +d)(x 2 +a)}=0 it further follows that: c=−Ex2Ex4−Ex6(Ex4)2−Ex4 d=(Ex4)2−Ex6Ex2(Ex2)2−Ex4

[0075] This results in the following according to the relations given above in (3): Ex2=1+σ2 Ex4=4125+6σ2+3σ4 Ex6=7325+61525σ2+45σ4+15σ6

[0076] In the equalizer 100, the kernels defined in (1) are transformed by the polynomial processing unit 102, which operates based on the polynomials and coefficients determined above. The output vector q after the polynomial processing in the processing unit 102 is (for the third order, ie, M=3), for example q(k)=[x(k),x(k−1),x(k−2),x2(k)+a,x(k)x(k−1),x(k)x(k−2),x2(k−1)+a,x(k−1)x(k−2),x2(k−2 )+a,x3(k)+bx(k),(x2(k)+a)x(k−1),(x2(k)+a)x(k−2),…,(x2(k−1)+a)x(k−2),x3(k−2)+bx(k−2)]T

[0077] The diagonal autocorrelation matrix is ​​then given by R qq The determination of the inverse matrix is ​​trivially easy from the diagonal vector of R qq , That means diag(Rqq)=[v1,v1,v1,v2,v3,v3,v2,v3,v2,v4,v5,v5,v5,v6,v5,v4,v5,v5,v4]T v1=Ex2, v2=Ex4−(Ex2)2, v3=(Ex2)2, v4=Ex6−(Ex4)2 / Ex2 v5=Ex4Ex2−(Ex2)3,v6=(Ex2)3

[0078] Now a new vector can be defined: z=1 / diag(Rqq)=[z1,z2,…,z19]

[0079] Thus, equation (2) can be simplified and the update algorithm is as follows: wj(k+1)=wj(k)+μekzjqj, j=1,2,…,19 wdc(k+1)=wdc(k)+μek

[0080] In the equalizer 100, each core 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 shown in Fig. Figure 4 shows the ACM after polynomial processing (here, M=3). The µ-working range can be adjusted to specific cases.

[0081] A simplified Volterra filter uses pe k x j for the coefficient update, whereas the Wiener equalizer updates the coefficients using pe k z j q j updated. The results of the multiplication µz jcan be stored, for example, in a lookup table, so that they do not introduce any noticeable complexity. Also, the number of multiplications is identical in both cases (there is only a different number of additions), and it can be concluded that these two approaches have a similar complexity of O(M 3 ) have.

[0082] At lower BERs, the parameter σ can be freely set to zero, while setting σ to larger values ​​improves stability and accelerates acquisition at higher BERs (ISI and noise scenario). The diagonal vectors for M=[35,5,7] (i.e., 35 linear coefficients, five second-order coefficients, and seven third-order coefficients) and σ=0 and 125 are in Fig. 5 (for the PAM-4 levels ±15, ±35 ).

[0083] The noise variance influences the update algorithm, and σ can be estimated and adjusted correctly. The use of σ=125 offers suboptimal but very good performance. The Equalizer 100 preferably starts with this value and then stabilizes the coefficients by estimating the noise. Initial acquisition may be somewhat slower than in the optimal case, when the noise standard deviation σ is better known, but after the acquisition phase, the channel may vary (PMD, power fluctuations, etc.), and the filter should track such fluctuations.

[0084] The tracking performance depends on the gradient factor µ and the standard deviation σ. The gradient factor µ is usually unchanged after the acquisition period. However, the tracking ability can be improved if the polynomial coefficients are calculated using the true value of σ. Therefore, the noise standard deviation σ - as already discussed with respect to Fig. 1 and Fig. 2 - is estimated on the equalized signal y after the filter 101 by the estimation unit 103. 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 using σ=14∑i=03σi where σ i the noise standard deviation of an i-th PAM-4 level L i which is given by the formula Li=−35+2i5 is defined, with i = 0, 1, 2, 3.

[0085] Five Borders B i and B i+1 are introduced and replaced by the formula Bi=−45+2i5 defined for i = 0, 1, 2, 3.

[0086] Records within the boundaries are labeled S i denoted by i = 0, 1, 2, 3. The mean and standard deviation for each PAM-4 level are easily estimated. Means are accurately estimated, while deviations are underestimated (smaller than they are). Therefore, σ is preferably increased in small steps, and the best Gaussian fit is preferably determined using histograms—as indicated by the symbols in Fig. 6. The best fit uses the mean square deviation to capture the best σ value. That is, the estimator 103 is designed to use the mean square deviation to determine the best Gaussian fit. To obtain histograms, only 11 binary values ​​are sufficient (see the cycles in Fig. 6). After adjusting the PDF, 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 best.

[0087] When channel conditions are extreme (noise, distortion, clock jitter, etc.), increasing the number of linear taps can lead to poor acquisition. The coefficients at the ends may not converge, resulting in suboptimal performance or even a complete acquisition failure. In most cases, noise is the primary cause of such behavior. However, the same behavior has been observed in experiments, not only in noise-dominated scenarios but also in highly nonlinear systems with pre- and post-tap values ​​that do not decay quasi-exponentially. This problem readily arises even in slightly distorted channels (low noise) when the number of taps is set to a very high value (e.g., 1000).

[0088] To solve this problem, the following procedure is used for the DD-LMS algorithm of the Fig. 3 shown LMS unit 302. 1. The signal power is normalized to 1. 2. The PAM-4 levels are −35,−15,+15,−, and +35 set. 3. All coefficients are set to 0 and the central tap value is set to 1. 4. Only N of M linear coefficients are updated, where 3 ≤ N ≤ 7 and 5×10 -4 < µ ≤ 10 -3 . 5. After K updates (103 ≤ K ≤ 104) all coefficients are set to 10 -4 ≤ µ < 5×10 -4 updated. 6. If the channel is static, µ < 10 -4 set.

[0089] The above procedure allows for correct detection with a high probability.

[0090] An experiment with 56 GBaud was carried out using high-bandwidth components constructed in Fig. 7 to obtain 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 into PAM-4 symbols. After signal amplification by a modulator driver (MD), the TOSA outputs an optical PAM-4 signal, which is optically amplified before a ROSA acquires the optical signal. The electrical PAM-4 signal is electrically amplified and acquired over the real-time domain. Since the transmitter and the real-time domain clock sources were not synchronized, clock recovery based on a Mueller and Mueller phase detector (M&MPD) in a second-order phase-locked loop (PLL) with a bandwidth of 4 MHz was used to compensate for the frequency offset and timing jitter. The signal is equalized, and the Gray BER is estimated.

[0091] Despite the high-bandwidth components used in the experiment, some nonlinearities arise. These nonlinearities are Fig. 8 can be seen. Fig. Figure 8 shows in (a) the averaged eye diagram over three symbol intervals at an input power of 0 dBm, after a linear equalizer with 55 taps and one sample per symbol.

[0092] Volterra equalizers were tested up to the fourth order with parameters M=[M1, M2, M3, M4]=[35, 7, 7, 7]. The equalization results are shown in Fig. 8 shown in (b), where V n an n-th-order Volterra filter. It can be seen that the third-order kernels provide higher gain at lower BER values, whereas the fourth-order kernels are irrelevant for equalization performance.

[0093] The development of the tap values ​​for Volterra and Wiener filters for u=10-4, M=[35,5,7], and Pin=-3 dBm (σ was 125 was monitored. Initially, all coefficients were set to 0 and w1(17) was set to 1. The evolution of the tap values ​​is very specific and unstable for a conventional Volterra equalizer, and this evolution in Fig. 9 (a), which shows the third-order kernels. Even after two million updates (symbols), the third-order kernels are not well settled and are likely causing instabilities in the first-order kernels. On the other hand, the equalizer 100 is after 5×10 5 Symbols are very stable, as in Fig. 9 (b), which shows the evolution of the third-order kernels. Compared to the 100-degree equalizer, the conventional Volterra equalizer requires about four times as much time to stabilize the tap values. It is interesting to note that both equalizers are stable after 10 6Symbols exhibit similar BER behavior. At lower BERs, the parameter σ can be freely set to zero, while setting σ to larger values ​​improves stability and accelerates acquisition at higher BERs (ISI and noise scenario).

[0094] The second-order coefficients are not affected. However, the evolution of the first-order coefficients is similar to that of the third-order kernels. For the Fig. 10 Volterra equalizer shown in (a) requires much more acquisition time than the one shown in Fig. 10 equalizer 100 shown in (b).

[0095] The present invention has been described in connection with various embodiments by way of example and implementation. However, from an examination of the drawings, this disclosure, and the independent claims, further variations may be deduced and practiced by those skilled in the art who practice the claimed invention. In both the claims and 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 perform the functions of multiple entities or items recited in the claims.The mere fact that certain measures are recited in the various interdependent claims does not indicate that a combination of those measures cannot be used in an advantageous implementation.

Claims

[1] Equalizer (100) for a four-stage pulse amplitude modulation signal, PAM-4 signal, the equalizer (100) comprising: a filter (101), a processing unit (102) designed to apply polynomial processing to an input signal x of the equalizer in order to output a processed signal q to the filter (101), an estimation unit (103) designed to estimate a noise standard deviation σ of a noise at an output signal y of the filter (101), and a calculation unit (104) designed to calculate polynomial coefficients based on the estimated noise standard deviation σ, wherein the polynomial processing in the processing unit (102) is based on orthogonal polynomials and on the calculated polynomial coefficients. [2] Equalizer (100) according to claim 1, wherein the filter is a Volterra filter. [3] Equalizer (100) according to claim 1 or 2, wherein the estimation unit (103) is designed to calculate the noise standard deviation σ according to the formula σ=14∑i=03σi to estimate, where σ i the noise standard deviation of an i-th PAM-4 level L i which is given by the formula Li=−35+2i5 is defined, with i = 0, 1, 2, 3. [4] Equalizer (100) according to claim 3, wherein the estimation unit (103) is designed to determine the noise standard deviation σ i the i-th PAM-4 level L i by finding the best Gaussian fit of a histogram of a obtained data set S i is determined. [5] Equalizer (100) according to claim 4, wherein the estimation unit (103) is designed to process the data set S i within limits B i and B i+1 which is given by the formula Bi=−45+2i5 are defined for i = 0, 1, 2, 3. [6] Equalizer (100) according to claim 4 or 5, wherein the estimation unit (103) is designed to use the mean square deviation to determine the best Gaussian fit. [7] Equalizer (100) according to one of claims 1 to 6, wherein the processing unit (102) is designed to set the noise standard deviation σ at the beginning of the polynomial processing to a starting value σ0, where σ0=125 [8] Equalizer (100) according to one of claims 1 to 7, wherein the highest coefficient of any orthogonal polynomial is 1. [9] Equalizer (100) according to one of claims 1 to 8, wherein a mean value of each polynomial is 0, and all polynomials are orthogonal. [10] Equalizer (100) according to one of claims 1 to 9, wherein the calculation unit (104) is designed to calculate the polynomial coefficients according to the formulas E[pi(x)pj(x)]=0 for i ≠ j, and E[pi(x)]=0 for i ≠ 0, where pk(x)=akkxk+ak−1kxk−1+…+a1kx1+a0k for k = 0, 1, ..., n. [11] Equalizer (100) according to one of claims 1 to 10, wherein the orthogonal polynomials the following polynomials G0(x)=1 G1(x)=x G2(x)=x2+a G3(x)=x3+bx G4(x)=x4+cx2+d wherein a, b, c and d are polynomial coefficients calculated by the calculation unit (104). [12] Equalizer (100) according to claim 11, wherein the calculation unit (104) is designed to calculate the coefficients a, b, c and d according to a=−Ex2 b=−Ex4Ex2 c=−Ex2Ex4−Ex6(Ex2)2−Ex4 d=(Ex4)2−Ex6Ex2(Ex2)2−Ex4 to calculate with Ex2=1+σ2 Ex4=4125+6σ2+3σ4 Ex6=7325+61525σ2+45σ4+15σ6. [13] Equalizer (100) according to one of claims 1 to 10, wherein for a third-order equalizer the processing unit (102) is designed to apply the polynomial processing to an input signal x of the equalizer which is calculated according to the formula x(k)=[x(k),x(k−1),x(k−2),x2(k),x(k)x(k−1),x(k)x(k−2),x2(k−1),x(k−1)x (k−2),x2(k−2),x3(k),x2(k)x(k−1),x2(k)x(k−2),…,x2(k−1)x(k−2),x3(k−2)]T is defined to output a processed signal q which is expressed according to the following formula q(k)=[x(k),x(k−1),x(k−2),x2(k)+a,x(k)x(k−1),x(k)x(k−2),x2(k−1)+a,x(k−1)x(k−2),x2(k−2 )+a,x3(k)+bx(k),(x2(k)+a)x(k−1),(x2(k)+a)x(k−2),…,(x2(k−1)+a)x(k−2),x3(k−2)+bx(k−2)]T is defined, where a and b are polynomial coefficients calculated by the calculation unit (104). [14] A method (200) for equalizing a four-level pulse amplitude modulation signal, PAM-4 signal, the method (200) comprising: Applying a polynomial processing (201) to an input signal x to output a processed signal q to a filter (101), Estimating (202) a noise standard deviation σ of a noise at an output signal y of the filter (101), and Calculating (203) polynomial coefficients based on the estimated noise standard deviation σ, where the polynomial processing (201) is based on orthogonal polynomials and on the calculated polynomial coefficients. [15] A computer program product comprising program code for executing, when run on a computer, the method (200) of claim 14.

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