Patent Publication Number: US-2022239510-A1

Title: Ethernet physical layer transceiver with non-linear neural network equalizers

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This disclosure claims the benefit of copending, commonly-assigned U.S. Provisional Patent Application No. 63/141,460, filed Jan. 25, 2021, which is hereby incorporated by reference herein in its entirety. 
    
    
     FIELD OF USE 
     This disclosure relates to the use of non-linear equalizers in a physical layer transceiver. More particularly, this disclosure relates to the use of non-linear neural-network equalizers in the transmit and receive paths of a physical layer transceiver such as an Ethernet physical layer transceiver, as well as for cancellation echo, near-end crosstalk, and far-end crosstalk. 
     BACKGROUND 
     The background description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the inventors hereof, to the extent the work is described in this background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted to be prior art against the subject matter of the present disclosure. 
     Many integrated circuit devices, particularly “systems-on-chip” (SoCs), include high-speed serial links between various device components (such as the individual silicon dice in an SoC). Typical high-speed serial links of that type, commonly known as “SERDES” (serializer/deserializer), may suffer from significant non-linearity or channel impairment in the signal path, as a result of, e.g., insertion loss, inter-symbol-interference (ISI), and, in an optical system, non-linearities such as dispersion loss, or, in a copper (i.e., wired) system, cross-talk, jitter, etc. Various forms of linear equalization typically are used, at the receiver end of such links, to attempt to mitigate such channel impairments. However, linear equalization may not be sufficient to compensate for such non-linearities, particularly when the signal levels (e.g., voltage levels) to be distinguished in a data signal are close together and there is a low signal-to-noise ratio (SNR). 
     SUMMARY 
     In accordance with implementations of the subject matter of this disclosure, a physical layer transceiver for connecting a host device to a wireline channel medium includes a host interface for coupling to the host device, a line interface for coupling to the wireline channel medium, a transmit path operatively coupled to the host interface and the line interface, including circuitry for encoding host data and driving encoded host data onto the wireline channel medium, a receive path operatively coupled to the line interface and the host interface, including circuitry for decoding data received from the wireline channel medium and passing the decoded data to the host interface, and adaptive filter circuitry operatively coupled to at least one of the transmit path and the receive path for filtering signals on the at least one of the transmit path and the receive path, the adaptive filter circuitry comprising a non-linear equalizer. 
     In a first implementation of such a physical layer transceiver, the adaptive filter circuitry may include a non-linear equalizer inline in the transmit path and configured to equalize transmit signals. 
     In a second implementation of such a physical layer transceiver, the adaptive filter circuitry may include a non-linear equalizer inline in the receive path and configured to equalize received signals. 
     In a third implementation of the subject matter of this disclosure, the adaptive filter circuitry may include non-linear echo cancellation circuitry coupled to both the transmit path and the receive path and configured to cancel echo between the transmit path and the receive path. 
     According to a first aspect of that third implementation, the adaptive filter circuitry may include non-linear echo cancellation circuitry operating in an analog domain of the physical layer transceiver. 
     According to a second aspect of that third implementation, the adaptive filter circuitry may include non-linear echo cancellation circuitry operating in a digital domain of the physical layer transceiver. 
     According to a fourth aspect of that third implementation, the adaptive filter circuitry may include non-linear crosstalk cancellation circuitry coupled to both the transmit path and the receive path for cancelling at least one of (a) near-end crosstalk, and (b) far-end crosstalk, between the transmit path and the receive path. 
     A fourth implementation of such a physical layer transceiver may further include adaptation circuitry configured to compare output of the adaptive filter circuitry to known data and to adapt the adaptive filter circuitry based on a cost function to reduce error in the output on a subsequent iteration. 
     In a fifth implementation of such a physical layer transceiver, the adaptation circuitry may be configured to adapt the adaptive filter circuitry based on cross-entropy between a respective bit and a log-likelihood ratio corresponding to the respective bit. 
     In a sixth implementation of such a physical layer transceiver, the non-linear equalizer may include a neural network equalizer. 
     According to a first aspect of that sixth implementation, the neural network equalizer may include a multi-layer perceptron neural network equalizer. 
     According to a second aspect of that sixth implementation, the neural network equalizer may include a radial-basis function neural network equalizer. 
     According to a third aspect of that sixth implementation, the neural network equalizer may be a reduced complexity neural network equalizer including a front-end filter having a first number of inputs and a second number of outputs, the second number being smaller than the first number, and a neural network filter having as inputs the outputs of the front-end filter. 
     In a first instance of that third aspect of the sixth implementation, the front-end filter of the reduced complexity neural network equalizer may include a finite-impulse-response filter to reduce the first number of inputs to the second number of inputs. 
     In a seventh implementation of such physical layer transceiver, the non-linear equalizer may include a linear filter and a non-linear activation function. 
     According to a first aspect of that seventh implementation, the non-linear activation function may be a hyperbolic tangent function. 
     According to a first aspect of that seventh implementation, the non-linear activation function may be a sigmoid function. 
     In accordance with implementations of the subject matter of this disclosure, a method of filtering interference in a physical layer transceiver for connecting a host device to a wireline channel medium includes performing non-linear equalization on at least one of the transmit path and the receive path for filtering signals on the at least one of the transmit path and the receive path, and adapting the non-linear equalizer based on cross-entropy between equalizer output and data signals on the wireline channel medium. 
     In a first implementation of such a method, performing non-linear equalization on at least one of the transmit path and the receive path may include performing non-linear equalization inline in the transmit path to equalize transmit signals. 
     In a second implementation of such a method, performing non-linear equalization on at least one of the transmit path and the receive path may include performing non-linear equalization inline in the receive path to equalize received signals. 
     In a third implementation of such a method, performing non-linear equalization may include performing non-linear echo cancellation between the transmit path and the receive path. 
     In a fourth implementation of such a method, performing non-linear equalization may include performing non-linear crosstalk cancellation for cancelling at least one of (a) near-end crosstalk, and (b) far-end crosstalk, between the transmit path and the receive path. 
     In a fifth implementation of such a method, performing non-linear equalization may include applying a non-linear activation function and performing linear filtering. 
     According to a first aspect of that fifth implementation, applying a non-linear activation function may include applying a hyperbolic tangent function. 
     According to a second aspect of that fifth implementation, applying a non-linear activation function may include applying a sigmoid function. 
     A sixth implementation of such a method may further include applying initial filtering of equalization inputs prior to performing the non-linear equalization, to reduce complexity by reducing number of inputs to the non-linear equalization. 
     According to a first aspect of that sixth implementation, applying initial filtering may include applying finite-impulse-response filtering. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Further features of the disclosure, its nature and various advantages, will be apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, in which like reference characters refer to like parts throughout, and in which: 
         FIG. 1  is a representation of a physical layer transceiver that may incorporate implementations of the subject matter of this disclosure; 
         FIG. 2  is a representation of a particular implementation of a physical layer transceiver incorporating the subject matter of this disclosure; 
         FIG. 3  is a plot of an exclusive-OR function in a Cartesian coordinate space illustrating a problem solved by implementations of the subject matter of this disclosure; 
         FIG. 4  is a plot of a transformation of the exclusive-OR function of  FIG. 2  into a different coordinate space illustrating a solution based on implementations of the subject matter of this disclosure; 
         FIG. 5  is a diagram of a first implementation of a non-linear equalizer that may be used in accordance with the subject matter of this disclosure; 
         FIG. 6  is a diagram of a second implementation of a non-linear equalizer incorporating the subject matter of this disclosure; 
         FIG. 7  is a diagram of a third implementation of a non-linear equalizer incorporating the subject matter of this disclosure; 
         FIG. 8  is a diagram of a fourth implementation of a non-linear equalizer incorporating the subject matter of this disclosure; 
         FIG. 9  is a diagram of a fifth implementation of a non-linear equalizer incorporating the subject matter of this disclosure; 
         FIG. 10  is a diagram of a sixth implementation of a non-linear equalizer incorporating the subject matter of this disclosure; 
         FIG. 11  shows a generic implementation of a class of reduced-complexity non-linear neural network filters in accordance with the subject matter of this disclosure; 
         FIG. 12  is a diagram of a first implementation of a non-linear equalizer in the class of reduced-complexity non-linear neural network filters shown generically in  FIG. 11 ; 
         FIG. 13  is a diagram of a second implementation of a non-linear equalizer in the class of reduced-complexity non-linear neural network filters shown generically in  FIG. 11 ; 
         FIG. 14  is a diagram of a third implementation of a non-linear equalizer in the class of reduced-complexity non-linear neural network filters shown generically in  FIG. 11 ; 
         FIG. 15  is a diagram of an alternative representation of the implementation of a reduced-complexity non-linear neural network filters shown in  FIG. 14 ; 
         FIG. 16  is a diagram of a fourth implementation of a non-linear equalizer in the class of reduced-complexity non-linear neural network filters shown generically in  FIG. 11 ; 
         FIG. 17  is a diagram of a fifth implementation of a non-linear equalizer in the class of reduced-complexity non-linear neural network filters shown generically in  FIG. 11 ; 
         FIG. 18  is a graphic representation of a non-linear function that may be equalized using the reduced-complexity non-linear neural network filter of  FIG. 17 ; 
         FIG. 19  is a diagram of a sixth implementation of a non-linear equalizer in the class of reduced-complexity non-linear neural network filters shown generically in  FIG. 11 ; 
         FIG. 20  is a flow diagram illustrating a method according to implementations of the subject matter of this disclosure; and 
         FIG. 21  is a flow diagram illustrating a portion of the method of  FIG. 20 . 
     
    
    
     DETAILED DESCRIPTION 
     As noted above, integrated circuit devices may include high-speed SERDES links between various device components. Typical SERDES links may suffer from significant non-linearity or channel impairment in the signal path, as a result of, e.g., insertion loss, inter-symbol-interference (ISI), and, in an optical system, non-linearities such as dispersion loss or, in a copper (i.e., wireline) system, cross-talk, jitter, etc. Various forms of linear equalization typically are used, at the receiver end of such links, to attempt to deal with such channel impairments. 
     However, particularly in an Ethernet physical layer transceiver (PHY), linear equalization may not be sufficient to compensate for such non-linearities, because the signal levels (e.g., voltage levels) to be distinguished in a data signal may be close together. For example, as opposed to typical non-return-to-zero (NRZ) signaling, which uses two levels to represent ‘0’ and ‘1’, a SERDES in an SoC device may use 4-level pulse-amplitude modulation (PAM4) signaling having four voltage levels, but with the same maximum voltage swing as NRZ signaling, to represent four possible two-bit symbols (‘00’, ‘01’, ‘10’, ‘11’). Moreover, Ethernet signaling may use an even higher modulation, such as 8-level pulse-amplitude modulation (PAM8) or 16-level pulse-amplitude modulation (PAM16) or higher. Thus, rather than one threshold within that voltage range dividing between two signal levels, there could be fifteen (or more) thresholds within the voltage range, dividing among as many as sixteen (or more) signal levels. Linear equalization may not be sufficient to correctly assign received samples near the thresholds between levels to the correct transmitted bit or symbol when the thresholds are close together that close together and the signal-to-noise ratio is low. 
     Moreover, in Ethernet-type signaling, there may be many signal sources on the channel contributing to various forms different of interference—particularly echoes, near-end crosstalk and far-end crosstalk. 
     In accordance with implementations of the subject matter of this disclosure, non-linear equalization is used to compensate for non-linearities in the PHY channel, as well as to cancel echoes, to cancel near-end crosstalk, and to cancel far-end crosstalk, thereby reducing the bit-error rate (BER). In different implementations, different types of non-linear equalizers may be used. 
     Conceptually, a linear equalizer performs the separation of samples for assignment to one level or another by effectively drawing a straight line between groups of samples plotted in a two-dimensional (e.g., (x, y)) space. In channels that are insufficiently linear, or where the levels are too close together, there may not be a straight line that can be drawn between samples from different levels on such a plot. A non-linear equalizer effectively re-maps the samples into a different, non-linear (e.g., radial or polar) space in which the samples from different levels may be separated by a straight line or other smooth curve. 
     A non-linear equalizer in accordance with implementations of the subject matter of this disclosure may be more or less complex. For example, a non-linear equalizer may have more or fewer variables, or taps, with complexity being proportional to the number of variables. In addition, a non-linear equalizer that operates at the bit level—i.e., operates separately on the bits of each symbol (e.g., two bits/symbol for PAM4 signaling) rather than on the symbol as a whole—may be less complex than a non-linear equalizer that operates at the symbol level. Either way, greater complexity yields greater performance when all other considerations are equal. However, greater complexity also may require greater device area and/or power consumption. 
     Types of non-linear equalizers that may be used in accordance with the subject matter of this disclosure may include multi-layer perceptron neural network (MLPNN) equalizers, and reduced-complexity multi-layer perceptron neural network (RC-MLPNN) equalizers, as well as radial-basis function neural network (RBFNN) equalizers, and reduced-complexity radial-basis function neural network (RC-RBFNN) equalizers, as described in more detail below. 
     Performance of the non-linear equalizer may be affected by the cost function used for adaptation of the equalizer. For example, according to implementations of the subject matter of this disclosure, the non-linear equalizer may use one of various different cost functions for adaptation, including either a minimum mean-square error (MMSE or MSE) cost function, or a cross-entropy (CE)-based cost function. A CE-based cost function may yield a better result than an MMSE cost function, but a CE-based cost function is more complex than an MMSE cost function. 
     Therefore, according to implementations of the subject matter of this disclosure, the choice of which form of non-linear equalizer to use, and of which cost function to use, may be a tradeoff of complexity (and therefore expense) versus performance. 
     The subject matter of this disclosure may be better understood by reference to  FIGS. 1-21 . 
       FIG. 1  illustrates the structure of a physical layer transceiver  100  that may incorporate implementations of the subject matter of this disclosure, in the context of a communications channel such as an Ethernet network channel. Physical layer transceiver  100  may include a transmitter path/channel  101 , and a receiver path/channel  102 , for data flowing between a host device  170  and wireline channel medium (e.g., cable)  180 . A host interface  171  couples transmitter path/channel  101  and receiver path/channel  102  to host device  170 , while a medium-dependent interface (MDI)  181  couples transmitter path/channel  101  and receiver path/channel  102  to channel medium  180 . 
       FIG. 2  shows the details of a particular implementation  190  of a physical layer transceiver  100  in accordance with the subject matter of this disclosure. In the transmitter path/channel  101  of transceiver  190 , an encoder  140 —e.g., a forward-error correction (FEC) encoder—can be used to encode transmit data bits  146  desired to be transmitted, followed by a pulse-shaping circuit  141  to manipulate time-domain characteristics of the transmit waveform so that signal timing information can be easily extracted on the receiver side. A transmit equalizer  142  may be provided to subtract out undesired components, or to recapture signal components that spread to neighboring symbols, before transmission. The equalized output is converted from digital to analog for transmission by transmit digital-to-analog converter  143 , and then driven onto channel medium  103  by transmit driver  144  via hybrid coupler and transformer  145  (serving as the MDI  181 ). 
     In the receiver path/channel  102  of transceiver  190 , data may be received from the channel medium  180  at the hybrid coupler and transformer  145 , and sent to an analog front end  151  of receiver  102 , and then to an analog-to-digital converter (ADC)  152 . An equalizer  153  can include one or more equalizers to remove interference. The output of the equalizer block  153  is sliced at slicer  154  and provided to a decoder  155 —e.g., a forward-error correction (FEC) decoder—which outputs received data bits  156 . 
     An analog echo canceller  161  may be provided between transmit path  101  and the analog domain of receive path  102  at  112 . A digital echo canceller  162  may be provided between transmit path  101  and the digital domain of receive path  102  at  122 . Crosstalk cancellers  163 , which may filter near-end crosstalk, far-end crosstalk or both, also may be provided between transmit path  101  and the digital domain of receive path  102  at  122 . 
     In accordance with implementations of the subject matter of this disclosure, any one or more of transmit equalizer  142 , receiver equalizer  153 , analog echo canceller  161 , digital echo canceller  162 , and crosstalk cancellers  163 , may be based on non-linear filters, and particularly on non-linear neural network filters. Suitable non-linear neural network filters are described in copending, commonly-assigned U.S. patent application Ser. No. 17/248,658, filed Feb. 2, 2021 and copending, commonly-assigned U.S. patent application Ser. No. ______, filed concurrently herewith (Attorney Docket No. MP13358/004048-0819-101), each of which is hereby incorporated herein by reference in its respective entirely. 
     An adaptation function  164  may compare log-likelihood ratios  165  output by equalizer  153  to output data bits  156  or, during a training mode, to training bits  166 , to adapt the various non-linear equalizers  142 ,  153 ,  161 ,  162 ,  163 . 
     The purpose of implementing equalization on the channel is to correct for various sources of interference referred to above and thereby effectively move samples that are on the wrong side of the threshold to the correct side of the threshold. Linear equalization effectively takes a plot of the samples in a two-dimensional (x, y) space and draws a straight line between the samples to indicate where the threshold ought to be. However, in a channel with non-linearities, there may be no straight line that can be drawn on that two-dimensional plot that would correctly separate the samples. In such a case, non-linear equalization can be used. Non-linear equalization may effectively remap the samples into a different space (e.g., having a different scale or coordinate system) in which there does exist a straight line that correctly separates the samples. 
     Alternatively, the non-linear equalization function may remap the samples into a space in which there exists some smooth curve other than a straight line that correctly separates the samples. For example, the non-linear equalization function may remap the samples into a polar-coordinate or radial space in which the samples are grouped into circular or annular bands that can be separated by circles or ellipses. 
     The advantage of non-linear equalization over linear equalization in a non-linear channel may be seen in a simplified illustration as shown in  FIGS. 3 and 4 , where the signal to be equalized is characterized by the exclusive-OR (XOR or CI) function.  FIG. 3  is plot of y=x 1 ⊕x 2  in (x 1 , x 2 ) space, where the open dots  201 ,  202  represent y=0 and cross-hatched dots  203 ,  204  represent y=1. It is apparent that there is no straight line that can be drawn separating the open dots from the cross-hatched dots. 
     However, a radial basis function 
     
       
         
           
             
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     can be used to transform the XOR function from the linear Cartesian (x 1 , x 2 ) space to a non-linear radial (φ(r 1 ), φ(r 2 )) space as follows: 
                                             x 1     x 2     φ(r 1 )   φ(r 2 )   y                                                    0   0   0.1353   1   0       0   1   0.3678   0.3678   1       1   0   0.3678   0.3678   1       1   1   1   0.1353   0                    
which is diagrammed in  FIG. 4 . As can be seen, when mapped into the non-linear radial (φ(r 1 ), (φ(r 2 )) space, the values  301 ,  302 ,  303  (as can be seen, both of the two y=1 points  201 ,  202  in (x 1 , x 2 ) space map to the same point  301  in (φ(r 1 ), φ(r 2 )) space) of the XOR function  300  may be separated by straight line  304 .
 
     As discussed below, various types of non-linear equalizers are available. Whatever type of non-linear equalizer is used may be adaptive to account for changing channel conditions. Various forms of cost function may be used for adaptation, to reduce errors on subsequent iterations. 
     One type of adaptation function that may be used is minimum mean-squared error (MMSE), where the mean-squared error (MSE) is defined as the square of the norm of the difference between the equalized signal (Y) and the ideal signal (Ŷ). The equalizer may initially be adapted in a training mode in which the ideal signal values are available. Later, during run-time operation, the detected output values of the equalized channel should be close enough to the ideal values to be used for adaptation. 
     Another type of adaptation function that may be used is the cross-entropy (CE) between a training bit and its log-likelihood ratio (LLR). In particular, cost function circuitry may be configured to compute a cross-entropy value indicative of a difference between a probability distribution of the detected bit value (which is a function of the LLR signal) and a probability distribution of the training bit value. The cost function circuitry then adapts the equalizer by setting an equalizer parameter (e.g., one or more coefficients of filter taps of the equalizer) to a value that corresponds to a minimum cross-entropy value from among the computed cross-entropy values and one or more previously computed cross-entropy values, to decrease a bit-error rate for the channel. As in the case of MSE equalization, the equalizer may initially be adapted in a training mode in which the ideal signal values are available. Later, during run-time operation, the detected output values of the equalized channel should be close enough to the ideal values to be used for adaptation. Specifically, if any forward error correction code (FEC) decoder (e.g., a Reed Solomon (RS) decoder or Low-Density Parity Check (LDPC) decoder) is available after the equalizer, then successfully decoded frames from the FEC decoder output may be used for adaptation. 
     LLR may be defined as the relationship between the probability (P 0 ) of a bit being ‘0’ and the probability (P 1 ) of a bit being ‘1’: 
     
       
         
           
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     The cross-entropy between a training bit and its LLR may be computed as follows: 
     
       
         
           
             
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     When the true bit is a logic ‘0’ but the probability of the detected bit represented by the LLR indicates that P 0 =0, or the true bit is a logic ‘1’ but the probability of the detected bit represented by the LLR indicates that P 1 =0, then the true value is the complete opposite of the expected value, meaning that cost (cross-entropy) approaches infinity. On the other hand, when the probability of a detected bit value as indicated by the LLR agrees with the true bit value, then cross-entropy equals zero. Insofar as in most cases both probabilities P 0  and P 1  are higher than 0 and lower than 1, cross-entropy will be a finite non-zero value. Thus, this cost function can be used for adaptation and reflects the quality of the detected bits, with the goal being to minimize cross-entropy. 
     The gradient of cross-entropy with respect to the LLR may be computed by substituting for P 0  and P 1  in the cross-entropy equation: 
     
       
         
           
             
               
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     The LLR may be adapted to minimize cross-entropy (i.e., 
     
       
         
           
             
               
                 
                   
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     as follows: 
         LLR   t+i   =LLR   t   −α·P   1  if bit=0 
         LLR   t+i   =LLR   t   +α·P   0  if bit=1 
     A negative LLR means bit=0 has a higher probability than bit=1, while a positive LLR means bit=1 has a higher probability than bit=0. In these equations, P 0  and P 1  are probabilities and therefore are positive values, and α is an adaptation bandwidth which also is positive. Therefore, when the true bit=0 then adaptation using cross-entropy will make a negative LLR more negative, and when the true bit=1 then adaptation using cross-entropy will make a positive LLR more positive. Therefore, cross-entropy-based adaptation maximizes the magnitude of the LLR and hence is a maximum-likelihood adaptation which reduces BER. Thus, adaptation of the equalizer to minimize cross-entropy also minimizes BER. 
     If one assumes that there is a general computation graph from parameter X→Y→LLR→CE such that parameter X affects the value of output Y which affects the LLR, from which the cross-entropy may be computed, then the cross-entropy gradient can be expressed in terms of other parameters: 
     
       
         
           
             
               
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     Therefore, any parameter can be adapted to minimize the cross-entropy. 
     One suitable implementation of a non-linear filter that may be used in accordance with the subject matter of this disclosure is non-linear equalizer  401 , seen in  FIG. 5 . Non-linear equalizer  401  is a multi-layer perceptron neural network  402  providing an equalized signal (Y)  411  from input digitized samples  421  that are delayed at  431  and combined in multi-layer perceptron neural network  402 . 
     As seen in  FIG. 5 , multi-layer perceptron neural network  441  includes at least one hidden layer  450  of hidden nodes  451 . In this drawing only one hidden layer  450  is shown, but a multi-layer perceptron neural network equalizer in accordance with implementations of the subject matter of this disclosure may have multiple hidden layers (not shown). Similarly, while  FIG. 5  shows four hidden nodes  451  in hidden layer  450 , each hidden layer in a multi-layer perceptron neural network equalizer in accordance with implementations of the subject matter of this disclosure may have more or fewer hidden nodes  451 , reflecting the number of parameters (filter tap coefficients). 
     Each hidden node  451  multiplies delayed samples  421  (to avoid crowding the drawing, only one of delays  431  is shown as being coupled to nodes  451 ; however, each delay  431  is coupled to nodes  451 ) by parameters (filter tap coefficients; not shown) and then sums (Σ) the filter taps. Each hidden node  451  then applies to its computed sum a non-linear activation function (e.g., a hyperbolic tangent activation function, tanh (ƒ), although other non-linear activation functions may be used), to generate a node output, which is then passed to the next layer, and so on. The final layer  452  does not include a non-linear activation function but simply sums its inputs. 
     Hidden nodes  451  may receive inputs not only from feed-forward delays  431 , but also from feed-back delays  461 , representing samples  460  of a fed-back prior symbol decision  412  of slicer  402 , which may be helpful in mitigating inter-symbol interference. 
     The aforementioned parameters of non-linear equalizer  401  are adapted based on the output Y. One approach for adapting the parameters of non-linear equalizer  401  is to compute at  472  the error (e) with respect to an ideal sample Ŷ derived from training symbols  469 . Minimization of the mean square error at  473  may then be used as the cost function to adapt the filter tap coefficients at nodes  451  as indicated at  471 . 
     As an alternative to multi-layer perceptron neural network  401 , an implementation  500  ( FIG. 6 ) according to the subject matter of this disclosure may include a reduced-complexity multi-layer perceptron neural network  501  operating on input digitized samples  521 . Reduced-complexity multi-layer perceptron neural network  501  includes two feed-forward filters  542 ,  543 , which may, e.g., be finite-impulse-response (FIR) filters. A slicer  502  provides an output decision Y  512  that is fed back through decision-feedback equalizer (DFE)  544  and combined with the output of second feed-forward (e.g., FIR) filter  543  to mitigate inter-symbol interference from a previous symbol. Reduced-complexity multi-layer perceptron neural network  501  resembles a linear-equalizer  560  (including FIR filter  543  and decision-feedback equalizer  544 ), having as its input the output of an additional feed-forward filter  542  to which a non-linear activation function  545  (e.g., a hyperbolic tangent activation function, tanh (ƒ), although other non-linear activation functions may be used) has been applied. 
     Similarly to the case of non-linear equalizer  401 , the parameters of non-linear equalizer  501  are adapted based on the output Y. One approach for adapting the parameters of non-linear equalizer  501  is to compute the error (e) with respect to an ideal sample Ŷ derived from training symbols  569 . Minimization of the mean square error at  573  is then used as the cost function to adapt the filter tap coefficients of FIR filters  542 ,  543  as indicated at  571 . 
     However, as described above, cross-entropy may serve as a more effective cost function for adapting the parameters of a non-linear equalizer to minimize BER. 
       FIG. 7  shows an implementation  600  of a non-linear equalizer according to the subject matter of this disclosure. Non-linear equalizer  601  is a multi-layer perceptron neural network  641  providing four separate equalized signals (Y ij ; i=0, 1; j=0, 1)  611  from input digitized samples  621  that are delayed at  631  and combined in multi-layer perceptron neural network  641 . A softmax function: 
     
       
         
           
             
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     implemented in circuitry  602  provides an output decision (sym)  612 , which is fed back (after conversion at  648  to a voltage—e.g., − 1  for ‘00’, −⅓ for ‘01’, +⅓ for ‘10’ and +1 for ‘11’, in the case of a 4-level signaling system such as PAM4) to multi-layer perceptron neural network  641  to mitigate inter-symbol interference from a previous symbol, and an output log-likelihood ratio (LLR sym )  622 . 
     As in the case of  FIG. 5 , multi-layer perceptron neural network  641  includes at least one hidden layer  650  of hidden nodes  651 . In this drawing only one hidden layer  650  is shown but a multi-layer perceptron neural network equalizer in accordance with implementations of the subject matter of this disclosure may have multiple hidden layers (not shown). Similarly, while  FIG. 7  shows four hidden nodes  651  in hidden layer  650 , each hidden layer in a multi-layer perceptron neural network equalizer in accordance with implementations of the subject matter of this disclosure may have more or fewer hidden nodes  651 , reflecting the number of parameters (filter tap coefficients). 
     Each hidden node  651  multiplies delayed samples (to avoid crowding the drawing, only one of delays  631  is shown as being coupled to nodes  651 ; however, each delay  631  is coupled to nodes  651 ) by parameters (filter tap coefficients; not shown) and then sums (Σ) the filter taps. Each hidden node  651  then applies to its computed sum a non-linear activation function (e.g., a hyperbolic tangent activation function, tanh (ƒ), although other non-linear activation functions may be used), to generate a node output, which is then passed to the next layer, and so on. The final layer  652  does not have non-linear activation function but simply sums its inputs separately for each of the four symbols. 
     Hidden nodes  651  receive inputs not only from feed-forward delays  631 , but also from feed-back delays  661 , representing samples of a fed-back prior symbol decision  660 , for mitigating inter-symbol interference. 
     Because equalizer  601  provides soft output in the form of an LLR, the output may be used with a further outer decoder (not shown), which may be a forward error-correcting (FEC) decoder such as a low-density parity check (LDPC) decoder or a Reed-Solomon decoder. 
     The aforementioned parameters of non-linear equalizer  601  may be adapted to minimize cross-entropy, using cross-entropy adaptation circuitry  670 , between a training symbol ( ) that is obtained by grouping training bits  671 , and output log-likelihood ratio (LLRs sym )  622 . Cross-entropy adaptation circuitry  670  is able to adjust parameters of non-linear equalizer  601 , at  680 , to minimize the cross-entropy between the training symbol ( ) and the probability of the detected symbol which is represented by LLR sym    622 . During run-time, output bits  690  of an outer decoder (such as a Forward Error Correcting, or FEC, decoder; not shown), but only from successfully decoded frames, may be used in place of training bits  671 . 
       FIG. 8  shows an implementation  700  according to the subject matter of this disclosure including a reduced-complexity multi-layer perceptron neural network  741 , coupled with a decision-feedback equalizer  742 , as well as log-likelihood ratio circuitry  743  that inputs an equalized signal (Y)  711  derived from input digitized samples  721 , and outputs a symbol decision (sym)  744  and a log-likelihood ratio (LLR sym )  745  of that symbol decision, based on target symbol values Ŷ 00 , Ŷ 01 , Ŷ 10 , Ŷ 11 . 
     Reduced-complexity multi-layer perceptron neural network  741  includes two feed-forward filters  746 ,  747 , which may, e.g., be finite impulse response (FIR) filters. A non-linear activation function  748  (e.g., a hyperbolic tangent activation function, tanh (ƒ)), although other non-linear activation functions may be used) is applied to the output of feed-forward filter  746  which is then input to feed-forward filter  747 . Symbol decision  744  is converted at  749  to a voltage for input to decision-feedback equalizer  742 , the output of which is combined at  750  with the output of feed-forward filter  747  to mitigate inter-symbol interference from a previous symbol, to yield equalized signal (Y)  711 . 
     The parameters of feed-forward filters  746 ,  747  may be adapted to minimize cross-entropy between output log-likelihood ratio (LLR sym )  745  and “true” symbols obtained from true bits which may be training bits or, during run-time, the output of a further outer decoder (not shown). Cross-entropy adaptation circuitry  760  has, as an input, the output log-likelihood ratio (LLR sym )  745 . In a training mode, cross-entropy adaptation circuitry  760  also has as inputs known training bits  761 , which serve as “true” bits which are then grouped to obtain true symbols. Cross-entropy adaptation circuitry  760  is able to adjust parameters of feed-forward filters  746 ,  747 , at  770 , by minimizing the cross-entropy between the training symbol obtained by grouping training bits ( ) and the probability of the detected symbol which is represented by output log-likelihood ratio (LLR sym )  745 . At run-time, output bits  790  of an outer decoder (such as an FEC decoder; not shown), but only from successfully decoded frames, may be used in place of training bits  761 . 
     Because a neural network equalizer is capable of decorrelating the bits of a multi-bit symbol, such as the two bits in a PAM4 symbol, a further implementation  800  according to the subject matter of this disclosure may be provided ( FIG. 9 ). Implementation  800  includes an MLPNN equalizer  841  similar to MLPNN equalizer  541  in that it includes at least one hidden layer  850  of hidden nodes  851  in which samples delayed at  831  (to avoid crowding the drawing, only one of delays  831  is shown as being coupled to nodes  851 ; however, each delay  831  is coupled to nodes  851 ) are multiplied by parameters (filter tap coefficients; not shown) and then the filter taps are summed (Σ). Each hidden node  851  then applies to its computed sum a non-linear activation function (e.g., a hyperbolic tangent activation function, tanh (ƒ), although other non-linear activation functions may be used), to generate a node output, which is then passed to the next layer, and so on. 
     MLPNN  841  differs from MLPNN  541  in that the final layer  852  includes two nodes  853 ,  854 , in which the inputs are not merely summed as in layer  552  of MLPNN  541 , but also have applied after summation a non-linear activation function, different from the non-linear activation function of nodes  851 , that decorrelate the two bits of each symbol, with each node  853 ,  854  providing one of the two bits. The non-linear activation function of each node  853 ,  854  may be, instead of a hyperbolic tangent activation function, a sigmoid function having a profile similar to that of tanh (ƒ), but ranging from 0 to +1 rather than from −1 to +1. 
     Node  853  provides a probability estimate  863  (p(bit msb )) for the most significant bit of the two bits in a symbol, and node  854  provides a probability estimate  864  (p(bit lsb )) for the least significant bit of the two bits of the symbol. The two probability estimates  863 ,  864  are then compared in slicers  855  to a threshold value of 0.5 to a obtain bit estimate (e.g., bit=0 if p&lt;0.5 and bit=1 if p≥0.5) for each bit in the symbol. 
     In an implementation in which the signaling includes more than four levels (e.g., PAM8 or PAM16), there would be more bits per symbol (e.g., 3 or 4 bits, respectively). In such a case, there would be a corresponding number of nodes rather than just two nodes  853 ,  854 . 
     At  856 , the separate bits are grouped back into a symbol, then fed back at  857  and converted to a corresponding voltage at  858  (e.g., −1 for ‘00’, −⅓ for ‘01’, +⅓ for ‘10’ and +1 for ‘11’, in a 4-level signaling system such as PAM4) for input to feed-back delays  861 , representing samples of a fed-back prior symbol decision relative to the next inputs from feed-forward delays  831 , for mitigating inter-symbol interference. 
     Because implementation  800  operates at the bit level rather than at the symbol level, cross-entropy adaptation circuitry  870  also operates at the bit level, determining the cross-entropy based on the separate bit-level probabilities  863 ,  864  and the training bits  871 , or at run-time, the output  890  of an outer decoder (such as an FEC decoder; not shown). 
     At the bit level, cross-entropy may be determined by first determining the log-likelihood ratios from the probability estimates as described above. Starting with the most significant bit, where P 0  is p(bit msb=0 ) and P 1  is p(bit msb=1 ), LLR(bit msb ) can be computed. CE(bit msb ) can then be computed from LLR(bit msb ) and the most significant bit of the training bits or the outer decoder bits. Then using p(bit lsb=0 ) as P 0  and p(bit lsb=1 ) and P 1 , LLR(bit lsb ) can be computed. CE(bit lsb ) can then be computed from LLR(bit lsb ) and the least significant bit of the training bits or the outer decoder bits. The bit level cross-entropy is the sum of CE(bit msb )+CE(bit lsb ). 
       FIG. 10  shows an implementation  900  according to the subject matter of this disclosure including a reduced-complexity multi-layer perceptron neural network  941  which can decorrelate the bits of a multi-bit symbol, such as the two bits in a PAM4 symbol, coupled with a respective decision-feedback equalizer  942 ,  952  for each respective bit. 
     Reduced-complexity multi-layer perceptron neural network  941  includes a first feed-forward filter  946 , which may, e.g., be a finite impulse response (FIR) filter. A non-linear activation function  945  (e.g., a hyperbolic tangent activation function, tanh (ƒ), although other non-linear activation functions may be used) is applied to the output of feed-forward filter  946  which is then input to a second feed-forward filter  947 , and in parallel to third feed-forward filter  957 . Each of feed-forward filters  947 ,  957  produces a respective equalized bit output Y msb    944 , and Y lsb    954 . 
     A respective non-linear activation function  961 ,  962 , different from non-linear activation function  945 , is applied to each respective equalized bit output Y msb    944 , and Y lsb    954 . Non-linear activation functions  961 ,  962  may be, instead of a hyperbolic tangent activation function, a sigmoid function having a profile similar to that of tanh (ƒ), but ranging from 0 to +1 rather than from −1 to +1. 
     Non-linear activation function  961  provides a probability estimate p(bit msb ) for the most significant bit of the two bits in a symbol, and non-linear activation function  962  provides a probability p(bit lsb ) estimate for the least significant bit of the two bits of the symbol. Each of the two probability estimates is then compared in a respective slicers  955 ,  956  to a threshold value of 0.5 to a obtain bit estimate (e.g., bit=0 if p&lt;0.5 and bit=1 if p≥0.5) for each bit in the symbol. 
     At  970 , the two bits are grouped into a symbol  971 , and then converted to a corresponding voltage at  972  (e.g., −1 for ‘00’, −⅓ for ‘01’, +⅓ for ‘9’ and +1 for ‘11’) for input to decision feed-back equalizer  942  in the most-significant-bit path, and to decision feed-back equalizer  952  in the least-significant-bit path. The output of each respective decision feed-back equalizer  942 ,  952  is combined at  943 ,  953 , respectively, with the output of respective feed-forward filter  947 ,  957  to mitigate inter-symbol interference from a previous symbol, to yield the respective equalized bit outputs Y msb    944 , and Y lsb    954  that are, as described above, input to non-linear activation functions  961 ,  962  to yield. 
     Cross-entropy may be determined, from p(bit msb ), p(bit msb ), and training bits  981  or outer decoder output  990 , in cross-entropy adaptation circuitry  980  by, as in the case of implementation  800 , first determining the log-likelihood ratios from the probability estimates as described above. Starting with the most significant bit, where P 0  is p(bit msb=0 ) and P 1  is p(bit msb=1 ), LLR(bit msb ) can be computed. CE(bit msb ) can then be computed from LLR(bit msb ) and the most significant bit of the training bits or the outer decoder bits. Then using p(bit lsb=0 ) as P 0  and p(bit lsb=1 ) and P 1 , LLR(bit lsb ) can be computed. CE(bit lsb ) can then be computed from LLR(bit lsb ) and the least significant bit of the training bits or the outer decoder bits. The bit level cross-entropy is the sum of CE (bit msb )+CE (bit lsb ). 
     A number of additional reduced-complexity implementations of non-linear neural network filters which may be used in accordance with the subject matter of this disclosure are illustrated in  FIGS. 11-19 . 
       FIG. 11  shows a general implementation  1000  of a reduced-complexity non-linear neural network filter  1001  in accordance with the subject matter of this disclosure for equalizing two sets of inputs  111 ,  121  from two signal sources on wireline medium  180  (as discussed below in connection with, e.g.,  FIG. 12 , this is only an illustration, and there may be any number—i.e., one or more—sets of input signals). Reduced-complexity non-linear neural network filter  1001  accepts inputs  111 ,  121  of a certain complexity, but initially filters inputs  111 ,  121  through a front-end filter  1002  to reduce the complexity of inputs  111 ,  121 , before filtering reduced-complexity inputs  1011 ,  1021  through non-linear filter circuitry  1003 . Reduction of the complexity of inputs  1011 ,  1021  allows a reduction in the complexity (as measured by dimensionality) of non-linear filter circuitry  1003 , therefore the complexity of non-linear neural network filter  1001 , without having to reduce the complexity of the inputs  111 ,  121  being filtered. 
     A first implementation of a reduced-complexity non-linear neural network filter  1100 , shown in  FIG. 12 , is based on a radial-basis function non-linear neural network filter  1101 , with a finite-impulse-response-(FIR)-based front-end filter  1102 . 
     In radial-basis function non-linear neural network filter  1101 , digital samples from two inputs  1111 ,  1121  are delayed by delay line  1131  and combined in radial-basis function non-linear neural network  1141 . As seen in  FIG. 12 , radial-basis function non-linear neural network  1141  includes at least one hidden layer  1150  of hidden nodes  1151 . Each hidden node  1151  operates on each delayed sample with a radial-basis function, but to avoid crowding the drawing only some delays in delay line  1131  are shown as being coupled to each hidden node  1151 . The outputs of hidden layer  1150  are combined (e.g., by addition) at  1152  to provide Y output  1103 . 
     Each sample input at  1111 ,  1121  adds a parameter or dimension to radial-basis function non-linear neural network filter  1101 , increasing filter complexity. In order to reduce the complexity of radial-basis function non-linear neural network filter  1101 , reduced-complexity non-linear neural network filter  1100  includes front-end filter  1102 , which combines some of the inputs from ADC outputs  111 ,  121  to provide a reduced number of inputs  1111 ,  1121  to radial-basis function non-linear neural network filter  1101 . As can be seen in  FIG. 12 , in this implementation, front-end filter  1102  uses FIR filtering (each line connecting a delay  1112  to sum  1122  represents multiplication of a sample by a coefficient (not shown) forming a filter tap, with the taps being summed at  1122 ) to combine, e.g., every four input samples from ADC outputs  111 ,  121  into one input sample  1111 ,  1121 , thereby allowing a reduction in the complexity (as measured by dimensionality) of radial-basis function non-linear neural network filter  1101 , and therefore the complexity of non-linear neural network filter  1100 , without having to reduce the complexity of the inputs  111 ,  121  being filtered. The unseen coefficients may be parameters that adapted with a back-propagation algorithm and, for example, may be derived from the equation set forth above in connection with the cross-entropy gradient 
     In the implementation of  FIG. 12 , each set of input samples  111 ,  121  is processed in a separate portion of delay line  1112 , and in a separate portion of delay line  1131 . In this implementation, with two sets of input samples, each delay line is divided into two segments. However, more generally, the number of segments corresponds to the number of input sets. Thus, for a single input set, there would be only one segment (i.e., the delay line would not be segmented) but if there were three inputs sets, the delay line may be divided into three segments, etc. 
     A second implementation  1200  of a reduced-complexity non-linear neural network filter, shown in  FIG. 13 , also is based on a radial basis filter neural network filter  1201 , with a finite-impulse-response-(FIR)-based front-end filter  1202 . As in the case of front-end filter  1102 , front-end filter  1202  uses FIR filtering (each line connecting a delay  1212  to radial-basis function  1250  represents multiplication of a sample by a coefficient (not shown; see discussion above in connection with  FIG. 12 ) forming a filter tap) to combine, e.g., every four input samples from ADC outputs  111 ,  121  into one input sample  1211 ,  1221 , thereby allowing a reduction in the complexity (as measured by dimensionality) of radial-basis function non-linear neural network filter  1201 , therefore the complexity of non-linear neural network filter  1200 , without having to reduce the complexity of the inputs  111 ,  121  being filtered. 
     However, in this implementation, rather than being summed, the taps of delay line  1212  are input directly to the hidden nodes  1250  of radial-basis function non-linear neural network filter stage  1201 , which in this implementation are upstream of delay line  1231 . 
     Once again, with inputs  111 ,  121  from two sources, half  1213  of delay line  1212  of front-end filter  1202  is devoted to input  111 , while half  1214  of delay line  1212  of front-end filter  1202  is devoted to input  121 , with one respective hidden node  1250  of radial-basis function non-linear neural network filter stage  1201  for each input source  111 ,  121 . The same is true of delay line  1231  within radial-basis function non-linear neural network filter stage  1201 , with separate halves  1232 ,  1233  of delay line  1231  devoted to inputs deriving separately from inputs  111 ,  121 . Here too, the delays  1231  form individual taps of a final FIR filter, which are combined at summation node  1241  to yield the output Y. 
     A third implementation of a reduced-complexity non-linear neural network filter  1300 , shown in  FIG. 14 , is based on a multilayer perceptron (MLP) non-linear neural network filter  1302 , with a finite-impulse-response-(FIR)-based front-end filter  1301 . 
     Typically, an MLP filter includes a delay line for input samples, followed by at least one hidden layer in which the samples are summed and then passed through a non-linear activation function such as, e.g., a hyperbolic tangent function tanh (ƒ), followed by a layer including one or more summations. 
     In finite-impulse-response-(FIR)-based front-end filter  1301 , delay line  1331  is divided into a first portion  1332  receiving inputs  111  and a second portion  1333  receiving inputs  121 . Each line connecting a delay  1312  to sum  1322  represents a multiplication of a sample by a coefficient (not shown; see discussion above in connection with  FIG. 12 ) forming a FIR filter tap. The taps are summed by the summation portion of each hidden node  1350 , which includes a summation function followed by a non-linear activation function which in this implementation is a tanh (ƒ) function. Although the hidden layer is shown as having only one hidden node  1350  for all of the inputs in each respective set of inputs  111 ,  121 , in other implementations (not shown) there may be multiple nodes  1350  for each set of inputs  111 ,  121 . In any event, a set of outputs  1311  is generated based on front-end filtering of inputs  111 , and another set of outputs  1321  is generated based on front-end filtering of inputs  121 . 
     In this implementation, the boundary between the front-end filter  1301  and the MLP non-linear neural network filter  1302  runs through the hidden layer of hidden nodes  1350 , but that is not necessarily the case in all implementations. 
     MLP non-linear neural network filter  1302  in this implementation includes a respective tanh (ƒ) non-linear activation function as part of each respective one of hidden nodes  1350  and a FIR filter formed by a delay line  1312  and a summation node  1322 . A portion  1351  of delay line  1312  receives output samples  1311  from front-end filter  1302 , while a portion  1352  of delay line  1312  receives output samples  1321  from front-end filter  1301 . Each line connecting a delay  1312  to sum  1322  represents a multiplication of a sample by a coefficient (not shown; see discussion above in connection with  FIG. 12 ) forming a FIR filter tap, and the taps are combined at summation node  1322  to yield the output Y. 
     Reduced-complexity non-linear neural network filter  1300  may be represented as an equivalent filter arrangement  1400 , shown in  FIG. 15 . Reduced-complexity non-linear neural network filter  1400  includes four FIR filters  1401 ,  1402 ,  1403 ,  1404 , and two non-linear activation functions  1405 ,  1406  (which may be respective tanh (ƒ) non-linear activation functions). 
     FIR filters  1401 ,  1402  form finite-impulse-response-(FIR)-based front-end filter  1410 , with FIR filter  1401  receiving inputs  111  while FIR filter  1402  receives inputs  121 . FIR filters  1403 ,  1404  and non-linear activation functions  1405 ,  1406  form reduced-complexity non-linear neural network  1420 . In reduced-complexity non-linear neural network  1420 , activation function  1405  receives the outputs of FIR filter  1401  and passes those outputs, after non-linear activation, to FIR filter  1403 , while activation function  1406  receives the outputs of FIR filter  1402  and passes those outputs, after non-linear activation, to FIR filter  1404 . The outputs of FIR filter  1403  and FIR filter  1404  are combined at summation node  1408  to yield the output Y. 
     Another implementation of a reduced-complexity non-linear neural network filter  1500 , shown in  FIG. 16 , also is based on a multilayer perceptron (MLP) non-linear neural network filter  1502 , with a finite-impulse-response-(FIR)-based front-end filter  1501 . In this implementation  1500 , finite-impulse-response-(FIR)-based front-end filter  1501  includes two FIR filters  1511 ,  1521 , each of which filters a respective set of inputs  111 ,  121 . The respective outputs of FIR filters  1511 ,  1521  are combined by summation node  1531 . 
     The outputs  1541  of finite-impulse-response-(FIR)-based front-end filter  1501  are then filtered by multilayer perceptron (MLP) non-linear neural network filter  1502 , which includes a non-linear activation function  1512  (which may be a tanh (ƒ) non-linear activation function), followed by FIR filter  1522 . 
     In a variation  1600  of reduced-complexity non-linear neural network filter  1500 , shown in  FIG. 17 , a scalable bypass path  1601  is provided around non-linear neural network filter  1502 . Scalable bypass path  1601  is controlled by a scaling factor g ( 1611 ). FIR filter  1522  inherently includes a similar scaling control. The provision of scalable bypass path  1601  allows several modes of operation. First, if g=0, reduced-complexity non-linear neural network filter  1600  operates identically to reduced-complexity non-linear neural network filter  1500 . Second, by setting g=1, and setting the scaling factor of FIR filter  1522  to  0 , reduced-complexity non-linear neural network filter  1600  operates as a linear filter. This linear mode may be used as a “jump start” mode while the non-linear portion of the filter is adapting. 
     In addition, a non-linear function  1700  (particularly one that is close to a linear function  1701 ) can be approximated as a series of linear functions  1702  of different slopes, as shown in  FIG. 18 . By varying g to vary the slopes, non-linear function  1700  can be filtered using mostly finite-impulse-response-(FIR)-based front-end filter  1501 , which is linear, with non-linear neural network filter  1502  correcting for the differences between the segmented linear approximation and the actual non-linear function. 
     A similar variation  1800 , based on reduced-complexity non-linear neural network filter  1400 , is shown in  FIG. 19 . A scalable bypass path  1801  is provided around non-linear neural network filter  1420 . Scalable bypass path  1801  is controlled by a scaling factor g ( 1811 ). FIR filters  1403 ,  1404  of non-linear neural network filter  1420  inherently include a similar scaling control. By controlling g at  1811 , non-linear neural network filter  1800  can be operated in various modes in a manner similar to non-linear neural network filter  1600 . 
     In each of the implementations shown, additional filter layers or stages may be added (not shown). For example, when nonlinearity in the channel is severe or interference length in the time domain is longer, then more than one nonlinear transformation may be needed to separate signals. Each nonlinear stage would transform its input to a different space at the output. After multiple transformations, a final space would result where signals can be then linearly separated. In the implementations of  FIGS. 15-17 and 19 , each additional layer might include an additional non-linear activation function followed by an additional FIR filter. Moreover, as discussed above in connection with the delay-line implementations, the delay line could be segmented into groups of delays corresponding to different groups of inputs. Similarly, in the activation-function-plus-FIR-filter implementations, additional parallel sets of activation functions and FIR filters can be provided, feeding the common summation node or nodes. 
     It can be shown that the various implementations of a reduced-complexity non-linear neural network filter shown above provide nearly as good performance as a non-reduced-complexity non-linear neural network filter, particularly when adapted using cross-entropy. However, the reduced complexity provides substantial savings in device area and power consumption. 
     A method  1900  according to implementations of the subject matter of this disclosure is diagrammed in  FIG. 20 . 
     Method  1900  begins at  1901  where non-linear equalization is performed on at least one of the transmit path and the receive path in a physical layer transceiver for connecting a host device to a wireline channel medium, for filtering signals on the at least one of the transmit path and the receive path. At  1902 , the non-linear equalization is adapted based on cross-entropy between equalizer output and data signals on the wireline channel medium, and method  1900  ends. However, as seen at  1903 , optionally (indicated by dashed lines), initial filtering—e.g., finite-impulse-response (FIR) filtering—may be applied prior to the non-linear equalization to reduce complexity of the non-linear equalization. 
     As seen in  FIG. 21 , one implementation of performing non-linear equalization at  1901  may include applying a non-linear activation function—e.g., a tanh (ƒ) function—at  2001 , and then performing linear equalization at  2002 . 
     Thus it is seen that a physical layer transceiver using non-linear neural-network equalizers in the transmit and/or receive paths, and/or for cancellation echo, near-end crosstalk, and far-end crosstalk, has been provided. 
     As used herein and in the claims which follow, the construction “one of A and B” shall mean “A or B.” 
     It is noted that the foregoing is only illustrative of the principles of the invention, and that the invention can be practiced by other than the described embodiments, which are presented for purposes of illustration and not of limitation, and the present invention is limited only by the claims which follow.