NeuralEQ: Neural-Network-Based Equalizer for High-Speed Wireline Communication
Authors: Hanseok Kim
Jae Hyung Ju
Hyun Seok Choi
Hyeri Roh
Woo-Seok Choi
Venue: Submitted to ICLR 2023 (2023)
Keywords: Forward-backward algorithm
Equalizer
Neural network
BER

Abstract: Rapid growth of ML applications demand high-performance computing systems to perform massive data processing. In such systems, I/O bandwidth must be scaled up to prevent any performance degradation due to the limited data transfer rates. To meet this demand, recently wireline communication started adopting PAM4 signaling and DSP-based equalizers. However, multi-level signaling and conventional equalizing techniques degrade the bit-error-rate (BER) performance significantly. To mitigate this problem, this paper proposes a novel neural network architecture that mimics the forward-backward algorithm estimating the posterior probabilities in Hidden Markov Models. The proposed neural network overcomes the existing equalizer performance such as feed-forward equalizers or decision-feedback equalizers, while reducing the complexity of the forward-backward algorithm.

Under review as a conference paper at ICLR 2023
NEURALEQ: FOR HIGH-SPEED WIRELINE COMMUNICATION
EQUALIZER
Anonymous authors Paper under double-blind review
ABSTRACT
Massive data processing is required by various applications such as ML, video streaming, cloud services, etc. In such systems, I/O bandwidth must be scaled up prevent any performance degradation due to the limited data transfer rates. To meet this demand, wireline communication recently started adopting PAM4 signaling and DSP-based equalizers. However, multi-level signaling and conven- tional equalizing techniques degrade the bit-error-rate (BER) performance signif- icantly. To mitigate this problem, this paper proposes novel neural network architecture that mimics the wwwkwdb algorithm estimating the posterior probabilities in Hidden Markov Models. The proposed neural network overcomes the existing equalizer performance, such as feed-forward equalizers or decision- feedback equalizers, while reducing the complexity of the hewarddbackward al- gorithm.
INTRODUCTION
Recent advances in MI/AI technologies have benefited our daily life by enabling services like searching, recommendation, and translation. The rapid growth of ML applications, increasing the amount of data and computation, has driven enormous demands on high-performance computing systems. Accordingly, the I/O bandwidth and computing power of such systems must be scaled up to support large data transactions between computing cores without experiencing performance degradation due to their limited data transfer rates. To meet such high bandwidth demand for high- performance computing systems, the PCIe standard is being developed up to 64 Gbps for the next generation, and data-rate rate of NVlink specializing GPU communication reaches 50 Gbps (Temuçin) et al. 202T). Data-rate of the ethernet protocol is also being developed up to 112 Gbps to handle the high network load.
Although demands for higher data-rate for wireline communication keep increasing, limited band- width of wireline channels poses problems data transmission. Different from wireless communi- cation, wireline channels are time-invan invariant.. In addition, pursuing low latency and good I/O energy efficiency, wireline communication has relied on simple modulation like PAM2 and simple equal- izers, followed by symbol-by-symby detection. However, data-rate is increasing a short channel causes severe signal distortion, making it difficult to restore data at the receiver. To increase the data-rate further, recently, wireline communication started adopting multi-level sig- naling like PAM4 and DSP-based equalizers along with analog-to-digital converters (ADCs) on the receive side. Although PAM4 signaling enables data-rate to reach over 100 Gbps, as (Cordova et al. 2021 de Abreu Farias Neto et al., 2020; Krupnik et al. 2020), due to the peak power con- straint. PAM4 degrades SNR and makes the signal more vulnerable to inter-symbol interference (ISI) caused by limited channel bandwidth, resulting in bit-error-rate (BER) degradation. Under these cireumstances, equalizers to compensate for the channel loss play a critical role in high-speed wireline communication.
rapidly even
Equalizers can primarily divided into a linear equalizer, e.g., feed-forward equalizer (FFE), and a nonlinear equalizer, e.g., decision-feedback equalizers (DFE). FFE is discrete-time finite-impulse- response (FIR) filter that boosts the input in the frequency range where channel loss high. is simple to implement, but since incoming noise is also amplified, symbol detection after FFE shows limited BER performance. On the other hand, DFE does not boost noise and shows better BER, but the timing constraint of the feedback loop makes it difficult design high speed.

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The -aprollol,n speculative, technique mitigates the timing constraint but increases hardware complexity significantly, especially in PAM4, resulting in a loss of energy efficiency. Moreover, DFE alone cannot remove the pre-cursor ISL. Therefore. high-speed wireline communication typically uses both FFE and DFE to improve BER (Dikhaminjia et al. 2018).
Recently, thanks the development of deep learning. receivers using neural networks, especially recurrent neural networks (RNNs), have been studied, (Ye et al. 2018: Zhou et al., 2019 Kechriotis
at. 1994] Kim al., 2020) Gomez Diaz et al. (022), showing excellent performance for sequence detection. However, for wireline communication, RNNs also have disadvantages that 1) high-speed implementation difficult due to the timing constraint (past computation results are required for the
current operation), and 2) sequence detection incurs large latency.
In this paper, we propose novel neural network architecture that is amenable to high-speed imple- mentation and shows better BER than existing equalizers. The proposed neural network employs seed-forward architecture without any daata-dependeec--ndduee efficiently pipelined hardware architecture for high-speed receivers. For better BER performance, the proposed network trained to mimic the forward-backward (FB) algorithm that has been used to estimate posterior probabilities of the states in Hidden Markov Models (HMMs), while reducing the computational complexity of the FB algorithm.
timing constraints, which allows
This paper makes the following contributions:
Inspired by the forward-backwart backward algorithm, a novel neural network, NeuralEQ, whose algorithm is proposed.
structure is similar to that of the
We verify that the computational complexity of the proposed NeuralEQ is lower than that of the
algorithm. While having much lower complexity, NeuralEQ shows superior BER performance compared to conventional equalizers.
More than half the weights of NeuralEQ can be pruned without performance degradation, which reduces the complexity significantly. Moreover, NeuralEQ is found to be robust against ISI variation.
2 BACKGROUND
The core idea of this paper is to develop a novel neural network architecture that mimics the forward- backward (FB) algorithm in order to overcome the existing equalizer (EQ) performance such as FFE and DFE while reducing the complexity of the FB algorithm. In the following, a brief description of the FB algorithm and how it can be applied to wireline communication will be provided.
2.1 FORWARD-BACKWAN ALGORITHM
The FB algorithm, which also known as BCJR (Bahl et al. 1974) or MAP decoder, is widely used in modern wireless communication with its excellent performance. However, since wireline commu- nication requires a much higher data rate, it is challenging to run computationally heavy algorithm such FB. would be helpful to look at the formula in detail to understand the complexity of the FB algorithm.
The FB algorithm infers the maximum posterior of all hidden states in hidden Markov model. Let St be a hidden state variable, which is an element of a set of hidden states S, and X be an observed sequence from time to Then the FB algorithm is summarized as follows.
7' = P(S", x2: 1:1 P(X [St), 1 S
(1)
P(S lit) is probability of hidden states with a sequence from time 1 to which refers to forward probability. And P(X'+1'' ( is a conditional probability of sequence occurs when the state at time is given, which refers to backward probability. Each forward and backward probability can be computed using dynamic programming with the following equations. Note that P(S St, X1# r1:t is replaced with af, P(X++1
++' s;) is replaced with 8 for simplicity and aja is the transition probability from j-th state to i-th state.
a = Σ
a
1 P
='Is" = 8i); 1-11= Σ P/ag: = =015' =s;)
(2)
JE|S
jE[S]
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Finally, element- wise product of forward and backward probability gives the posterior probability for a given sequence from to T. which expressed with y in Equation
2.2 APPLYING FORWARD-BACKWARI ALGORITIM TO WIRELINE COMMUNICATION WITH LOSSY CHANNNL
The FB algorithm can applied when wireline communication is modeled as HMM. To this end, four parameters in HMM should be defined: (1) the observed sequence, (2) the hidden states, (3) the transition probability, and (4) the conditional probability of observables given each hidden state. First, the observed sequence is defined as the channel output distorted by ISI. Typically wireline links are tested with random data patterns, so the channel input is assumed to be random sequence without any coding (e.g. convolutional code). Note that, due to the channel ISI, correlation between adjacent channel outputs becomes nonzero. Second, each hidden state s; is defined as the possible channel input sequence with the length of the number of ISI, or [SI], which determines the channel output without any noise. For example, in PAM4 signaling, there are 1 possible hidden states, each corresponding
vector z, E (--,-1// 1/3, 1) |tsil Third, each hidden state has four possible state transitions with equal probability 1/4. Finally, assuming an additive white Gaussian noise channel, the conditional probability distribution of the observable can be written as:
T
P(X* z/S* =;)
1
exp
Wi)3) where b
Z
T
h²)
(3)
V2T02
Note that h is ISI coefficients. F shows how PAM2 wireline communication can be modeled with HMM describing four parameters HMM in wireline communication.
Exploiting the FB algorithm in wireline communication results in better BER performance compared the conventional EQs such FFE and DFE demonstrated in Figure Tb In this simulation, PAM4 signaling is used with ISI=[1.0, 0.4, 0.2, 0.1], while the number of FFE and DFE taps are chosen 8 and respectively, which are sufficient to compensate for ISI. Although the performance of the FB algorithm is excellent, its computational complexity is too high. There have been efforts reduce the complexity for wireless communication (Wang et al.) [006 [Thangarajah et al. 201T Talakoub al. 2007), but it is still difficult to use it in high-speed wireline communication. The number of hidden states increases exponentially with the length of ISI, and the forward and backward computations increase proportionally to the number of hidden states. Recently, the speed of wireline communication has reached 100 Gbps, so very challenging to implement the FB algorithm to operate this speed, and power consumption is very high even if implemented. In this study, using a neural network, while reducing the complexity of the FB algorithm, an equalizer superior to conventional FQs in terms of BER performance is developed.
Channel ISI
50
P(X
10-1
10-2
gaus. noise
Sg
P(X(s.)
Hidden state
$2
P(X(sg)
FFE
Transition probabiliy
10-4
DFE
Conditional probebillty
FWDBWD
P(X)s.)
ooservation
P(X(s,)
SNR (dB)
(b)
Figure 1: (a) Applying HMM to wireline communication, (b) Performance comparison between FFE, DFE, and FB
3 PROPOSED NEURAL EQUALIZER
There would be many ways to create an equalizer using the neural network. The first method would be the fully connected structure (Zhou et al. 2020 Oheh & Hoydis 2017 Schaedler al.,

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(3)
()
Y1
(2)
Y2
Y3
O2
(8)
(2)
Yisi
151
1st
11
x
:
P(+15)
P( 1s)
P(* 15)
P( 1s)
Rinco
(1)
x(2)
argmax
(3)
(4)
(a)
(
80
FF
Fully contrected
MOD
x1)
(2)
(3)
(4)
argmax y
Figure 2: (a) Computational graph of FB algorithm, (b) Mapping FB to neural network: i) two con- secutive products is replaced with two fully connected layers, ii) computing probability is replaced with single perceptron, iii) two consecutive products to compute gamma is replaced with two fully connected layers, (c) Proposed neural network based equalizer, NeuralEQ. (Note that thick lines used to represent full connections between components.)
2019). The fully connected structure has the advantage of being able to mimic any function, but, as can be inferred from the name, it often has unnecessarily many connections to implement the target function. There is another way utilize RNN. Many recent works such Ye al. 2018 Zhou
et al. 2019 Kechriotis et al. 1994 Kim et al. 2020, Gomez Diaz et al. 2022 implement equalizer as a neural network, RNN often used because RNN has excellent performance
sequence detector, and equalizer can be seen as a kind of sequence detector. In fact, although previous studies have implemented equalizers that perform well through RNN, RNN has the following two problems. First, because RNN has recurrent structure that requires past results in the current operation, it is difficult to satisfy timing margin when designing hardware requiring high-speed operation, as in wireline communication. Second, because of the recursive structure, the same weight is repeatedly multiplied by the output, which causes gradient vanishing or expanding problems, making training difficult (Pascanu al. 2013). Therefore, in view of these drawbacks, we develop novel neural network architecture, neither the fully connected structure nor RNN, and the proposed architecture inspired by the FB algorithm.

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3.1 STRUCTURE DESCRIPTION
The proposed neural network is designed to have a similar structure to the FB algorithm. Figure 2a shows an illustration of the FB algorithm computation assuming that one symbol is decoded from four received values. As expressed in Equation [ a is composed of the multiplication and sum of af, azi, and P(æ[sss, which are expressed by line connected the triangle in Figure 2a 3 calculated in the backward path is also drawn in the same way, and only the direction is opposite. Finally, the part where a and are multiplied to obtain y and find the state with the highest value is shown in the middle of Figure 2a
As shown in Figure Za when recovering the target symbol given a sequence of channel output, values further away from the target need to go through more complex calculations compared to those closer. However, in case of fully connected neural nets, they have uniform depth for all inputs. Thus if they are used to emulate the FB algorithm, parameters would be wasted for the inputs close to the target symbol. Therefore the proposed neural network architecture reflects this imbalance in computing depths.
To replace each component of the Figure 2a with a neural network, we first note that, in the part af = -jjesg a a,iP(="IS* si), ag. aji, and P(x)(s:) are multiplied in serial. P(r[s;) and az functions of observed sequence x, and ajz is constant given the system. We found that two fully connected layers are sufficient to learn this function. Then, the parts calculating P(x is replaced with a single perceptron. Finally,
replace the part Equation [ with two fully connected layers whose output layer has the size of the number of modulation level, |MOD]. For example, if PAM4 used, then the output layer has four neurons. These replacements are described in Figure 2b Note that the number of neurons per layer, which is expressed as N in Figure 2c] is not the same as [|] in the proposed architecture. Smaller N is desirable reduce the amount of computation, which will be described in more detail in Section 3.3
The main problem of implementing systems is that the recurrent structure incurs huge area and power overhead. On the other hand, the proposed neural network does not have any feedback structure, so efficient pipelining in imple- mentation is possible. DNNs composed of only fully connected layers also have the same benefits, but they are overparmeterized, causing a large computational and power overhead. Since the pro- posed architecture contains much less parameters obtain the same performance those with fully connected layers, can be implemented with less complexity.
equalizers in high-speed wireline communication
3.2 CONSIDERATIONS ON INPUT SIZE AND TARGET SYMBOL POSITION
In principle, the FB algorithm can decode the symbol at any location from input sequence. Even though Figure 2a describes the case when decoding the second symbol, it can calculate y's for any position between 1 and 4. However, it will not perform equally for all positions because the channel ISI can have both pre-cursors and post-cursors around the main-cursor, and for a given length of input sequence, as the target symbol position moves to the left (or right), the information on the pre- cursors (or post-cursors) disappears.
other words, position of the target symbol must be carefully chosen for good BER performance. In this study, we use the length of the input sequence and the target symbol position as 12 and 4, respectively. This generally showed good performance over various wireline channels tested. For the channels with much larger/less ISI compared to the tested ones, the input sequence length and target symbol position should be adjusted.
Let the channel input sequence be vector z, the signal with noise added to the channel output be vector x, and the channel impulse response be h, that is, x = z x h + n, where n ~ (O, I)) Also, the length of the input sequence is defined as T and the target symbol position is defined as D. the estimator 2 for the original signal z can be expressed as gt+D = Jw (xttt-1))) Note that fw indicates the function of proposed neural network, NeuralEQ. However, if there pre-cursors in the channel ISI, delay occurs between å and x, the final equation is summarized again as follows.
gt D lpre.cursorsl fw (I1
where |pre cursors] is the number of pre-cursors in ISI.
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3.3 COMPLEXITY
For the proposed neural network to be legitimate, the proposed NeuralEQ should be less complex than the original FB algorithm. To analyze complexity, we only compare the forward path equation of the FB algorithm and NeuralEQ. Even though both the FB algorithm and NeuralFQ consist of forward path, backward path, and gamma computation, forward and backward paths have identical structures and the computational load of gamma is ignorable. Analyzing results summarized as T
The computation type and number of operations required to calculate of from a E S), and at+1 from a E N) are shown in Table As shown in Figure 2b because one layer the FB algorithm is replaced with two layers of neural network, the forward-patt equation of NeuralEQ is also a two-layer computation. L.e. a' to a++1_ The number of multipliers of the FB algorithm, which is a dominant factor in computation complexity, is [S]? [S|. On the other hand, NeuralEQ has 2N² + N multipliers.
If N and S are the same, NeuralEQ has more multipliers, but it is not. [S] is determined by [
|SST|, where the length of ISI is 10 and the case of PAM4 is 4 200 However, Neu- ralEQ has a good performance for the length of ISI equals 10 even when N = 32. In this case, the number of multipliers is 2 + 32, which means NeuralEQ has 2 times fewer multipliers. In addition, we will discuss a method to further reduce complexity in Section 4.2 which shows further compression of the NeuralEQ's complexity.
Feature
FB algorithm
NeuralEQ
a = tanh Σ JEN oj Wif
forward-pali equation
+MWN+1r + 6))
af = Σ j
a P(x"ss))
D b (e() a = tanh(I jEN alw ++1 + b(t
where M = -tanhh ww (w'x""
# of multipliers
S
1.10 X 10
2N + N
2.08 x 10
# of adders
S
S
1.10 x 10
2N +N
2.08 x 10
# of conditional prob.
S
1.10 10
N/A
N/A
# of tanhs
N/A
N/A
3N
9.6 10
Table 1: Comparison of the number of computations between FB and NeuralEQ. The forward-path equation of NeuralEQ is derived for time:t+1 from time:t-1, unlike the FB algorithm. It is because one layer of the FB algorithm corresponds to two layers of NeuralEQ as shown in Figure 2 Also, notations referred by Figure 2 Then the numerical values were calculated for the case where PAM4 modulation is used, the length of ISI is 10 and N = 32.
3.4 TRAINING AND PERFORMANCE
The loss function of the proposed neural network defined as follows, using the notations in Equa- tion 4
loss
CrossEndropyyuw (T D-rtD-Dprr.uaaa
(5)
where N is the batchsize. general, the trained network may not be generalized well due to the lim- ited number of the training data. Even the proposed neural network shows low performance, if not provided various data patterns, the correlated ISI, and the various noise environments as examples. Therefore, securing a large number of training sets --:++-11 and for training. The good news is that in wireline communication, as long as data is continuously trans- mitted from the transmitter, is possible to generate x and z with infinite noise patterns and various ISI patterns. Because data could be infinitely generated, there was reason to repeatedly use the same training data. Thus we have no notion of 'epoch', or rather, the network was trained in a sin- gle epoch. Every time training data was needed, it freshly generated. Training, validation and evaluation sets were each randomly generated with the same ISI and SNR as parameters. The only difference is their size. Sequences of length 2e9, 2e8, 1e7 were used for training, validation, and
is advantageous

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evaluation respectively. Every time le7 training data was used, used le6 of validation data for validation. The training parameters are summarized in Table 2
To examine the performance of the proposed NeuralEQ, Figure B can be obtained with the same environment as set in Figure [b] i.c. [00[[1.000.40022.00111, the number of FFE taps is and the number of DFE taps is 3. It is shown that the neural network has superior performance and DFE, and has performance close to optimal decoder, the FB algorithm. In addition, results for various channel environments are confirmed in Section and all of them have better performance than conventional EQs.
FFE
10
Network parameters
Value
Input size
12
10-2
Target symbol position(D)
BER
# of neurons per layer(N)
Training parameters
Value
10
FFE
# of train, valid, eval sets Batchsize
2e9, 2c8, le7 8192 Cross Entropy Adam
DFE
10
FWDBWD
NEO
Loss function
Optimizer
14
Learning Rate
1e-3
SNR (dB)
Figure 3: Performance of NeuralEQ
Table 2: Proposed NeuralEQ syper-parammters
4 EXPERIMENT RESULTS
4.1 PERFORMANCE ON VARIOUS CHANNEL
Four different channels were prepared to evaluate the performance of the proposed NeuralE Each channel had -7dB -12dB, -6dB, and -21dB loss in Nyquist, respectively, and SBR was extracted from each channel and used for simulation. (refer to Appendix A In addition to NeuralEQ, the combination of FFE and DFE, which are conventional EQ, was also evaluated. Unlike the channels used in the section channels in this section have pre-cursors, which is more realistic, so it is challenging to perform properly by FFE or DFE alone, so they are used together and compared with NeuralEQ. In order to perform evaluation, the number of FFE taps is set to 24, and the number of DFE taps is 5, which is more than what is actually used in the PAM4 wireline application so that there is performance degradation due to lack of taps. The parameters of NeuralEQ are the same as those shown in Table except for the ISI of the 221 dB channel, which has worse ISI than other channels, so in this case, the length of the input sequence is increased from 12 to 24.
As illustrated in Figure 4 it can be shown that the performance of the proposed NeuralEQ in the whole SNR region is superior the combination of FFE and DFE with sufficient taps. In particu- lar, the gap in performance increases as the ISI, which show that NeuralEQ is more valuable than existing EQ in recent wireline communication that suffers more from high channel loss.
4.2 PRUNING
Although the proposed NeuralEQ needs multiple symbols per a single decoding, not all input sym- bols have the same information in decoding the target symbol. This is because the magnitude of the cursors decreases as the distance away from the main-cursor, so the information of the target symbol minor. On the other hand, the symbol at the main-cursor position has the most information. Due to this imbalance in information of symbol position, even if the complexity of the left and right side layers of the NeuralEQ is lower than those the middle, it can be expected that NeuralEQ will not significantly degrade performance.
The pruning @Janowsyy] 1989 Karnin 1990) is the efficient methodology to strike a balance between the difference of information and computational complexity, reducing unnecessary neurons. Pruning has been studied in a way that reduces hardware complexity without performance degradation from model that has been trained. In particular, (Frankle & Carbin 2019) consists that there exist
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109
10-1
10
10
10-2
10-3
BER
10-3
1074
10-4
10"*
10"1
NEO
NEO prune
10
10-6
10
20
25
15
SNR (dB)
SNR (dB)
(a)
(b)
10a
10
10
10
10"
10
BER
10"4
10-1
FFS+DFE
FFE+DFE
10-3
10-3
NEQ
NEQ
10
10
10
20
25
0
10
15 SNR (dB)
20
25
SNR (dB)
(c)
(d)
Figure 4: Performance comparison FFE+DFE, NeuralEQ, and NeuralEQ with pruning, for chan- nel loss (a) -7dB, (b) -12dB, (c), -16dB, and (d), -21dB. In all cases, NeuralEQ has better perfor- mance than FFE+DFE with or without pruning. Especially for channel loss -21dB, pruning increases the performance of NeuralEQ.
model which has better similar performance even after pruning, so we looked whether pruning can improve its performance even for the proposed NeuralEQ.
Figure 5 shows how much the performance degrades as the progress of pruning. Every pruning iteration, 10% of the total weight which are the smallest ones is pruned. The Y-axis is the normalized value of BER before pruning, and the x-axis represents sparsity. Performance degradation due to pruning appears differently for each channel. When the ISI is small, for -7dB lossy channel, performance degradation due to pruning small, and when the ISI large, relatively sensitive to pruning. Also, Appendix C describe how the sparisty of cach layer varies with the progress of pruning.
2.75
Loss:-7dB
Channel loss
of param.
# of param. after prune
2.50
Loss:-12dB
before prune
Diff.
Loss:-16dB
-7dB
38,564
13,446
-65%
LosB:-21dB
1.75
-12dB
38,564
16,600
-57%
1.50
-16dB
38,564
18,445
-52%
1.25
1.00
-21dB
76,964
36,812
-52%
10
20
30
40
50
60
70
80
90
Sparsity(%)
Table 3:
NeuralEQ parameter reduction after pruning for various channel
Figure 5: Normlized BER via sparsity
In Figure the performance is reevaluated using the pruned model. It can be that there is no significant difference between the performance before pruning or that it has better performance for the -21 dB channel. The number of parameters before and after Pruning is summarized in
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109
10-1
10
10-2
10-2
10-3
BER
10-1
ITCHOTE
10"4
HRE- 19 FEE-DFE,
10-4
PRE-DFS,
FEE-DrE
10"^
10"1
NEO,
WFO.
NFO,
10-6
10-a
10 SNR (dB)
20
25
10
15
SNR (dB)
(a)
(b)
10a
10a
10
10
10
10
10
BER
* HEIGHE
MEIGIE
10"6
FFE
10
FFE-DRE,
10-3
NEG, 1%
BEQ
10
NED. 1%
KED, 254
NEO. 27
10
10
10
20
25
0
10
15
20
25
SNR (dB)
SNR (dB) (d)
(c)
Figure 6: Performance variation due to ISI skew for channel loss (a) -ddB, (bb)-12dB,, (6) and (d) -2dBB which occurs in practical application. Both conventional EQ and NeuralEQ evaluated for comparison, and the magnitude skew of ISI is varied with 0%, 1%, and 2%. Regardless of ISI skew, NeuralEQ has better performance than conventional EQ.
Table Accordingly, pruning is a well-suited methodology to reduce number of parameters needed adaptively for each channel without degradation.
4.3 ROBUSTNESS TO ISI VARIATION
In practice, the channel environment during training and after training could be different. Therefore, test sets which has skew on ISI during training are generated and then we evaluate the performance degradation. The combination of FFE and DFE is also evaluated in the same environment control group. The amount of skew to ISI is as follows. When the single bit response of the channel is and let skew be s, hakew h where A o² I²I), and max(h)p (p {0, 0.01, 0.02)), which is proportional to the magnitude of main-cursor.
Figure shows how much the performance conventional EQ and NeuralEQ degrades when the skew is generated to ISI for each channel. In the case of 7 dB, skew does not affect performance much, but can be seen that the degree degradation increases as channel loss increases. It may not be deemed that NeuralEQ is stronger against ISI skew than conventional EQ, but it is not more sensitive. The performance superiority of traditional EQ and NeuralEQ is not reversed despite skew, and it can be confirmed that NeuralEQ has acceptable tolerance to ISI variation.
5 CONCLUSION
In this paper, we developed a neural-network-baar -ased equalizer suitable for high-speed wireline com- munication. was shown that the proposed neural EQ mimics the FB algorithm, reduces compu- tational complexity, and has superior performance to the existing equalizer. In addition, since there no feedback loop, the high-speed design is easier than the existing equalizer, such as DFE. The pruning technique more efficient hardware implementation. Also, we verified robustness to ISI variation is acceptable compared to the existing EQ.
applied to further reduee complexity without performance degradation for

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doi: 10.1 10.100991...2002229991 29010088...

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Under review as a conference paper at ICLR 2023
A CHANNELS FOR SIMULATION
The channels for performance evaluation of the proposed NeuralEQ are extracted from real PCB striplines with different lengths. Channels of loss 7dB, 12dB, 16dB, 21dB are from PCB stripline of length 3inch, 5inch, 7inch, 9inch respectively. The characteristics of the channels are shown in F Each channel has a loss between 7 dB and 21 dB at nyquist Trequency(Figure/Ze and the SBR of each channel is shown in Figure 7b The SBR of the channel extracted values only if it is greater than 0,001 times of the main-cursor, otherwise, it is forced to zero. The higher loss channel, the longer ISI it has.
2.59
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Figure 7: (a) Frequency response, and (b) single bit response of tested channels
B DECISION OF SNR FOR TRAINING
When creating a data set for training, must consider the amount of noise applied to the data. If noise is not applied at all, the neural network has not learned about the input sequence with noise, so noise will not be effectively removed during decoding. If the noise of the training set too large, it is highly likely that the network will not converge to the optimal point.
Figure 8 shows how the performance changes by changing the noise applied during training, that is. SNR. As expected, too large or too small noise degraded performance, and when trained for SNR with a BER of about 1e-2 empirically, a neural network with good overall performance was obtained.
should be noted that the noise applied to the test set and the noise applied the training set does not produce good performance. That is, the SNR used during training to have good performance does not match the SNR to be applied during the test. For example, the performance at 18dB SNR better when trained with 12dB SNR than the result of training at 18dB SNR in Figure Sd Simulation results for various channels show that training SNR having performance around le-2 BER used in training phase, the network is learned to have better performance in full range of SNR, and in this paper, these SNRs were found and trained for each channel.
C SPARSITY OF EACH LAYER WITH PRUNING PROGRESS
When pruning is performed on NeuralEQ, having small weight, low importance neurons, will be removed and high importance neurons will remain. Since ISI usually has smaller cursor value it goes toward the edge around the main-cursor, it can be assumed that the layers at the edges are less important than the middle even in NeuralEQ.
Figure 9 shows the NeuralEQ pruning for -dB, -22BB - 16dB, and -21dB channels and the sparsity of each layer for each pruning. The difference in sparsity between layers means the information
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per symbol is not important with the same amount for decoding. Pruning progress unimportant neurons efficiently.
D ROBUSTNESS SIGNAL QUANTIZATION
The quantization of channel signals must be preceded to implement the proposed NeuralBQ into dig- ital hardware. For such implementation to feasible, must be possible to limit the performance degradation due quantized inputs, while using reasonable number of bits for quantization. We anticipated the degradation, and the appropriate number bits, by training and evaluating the Neu- ralEQ with quantized channel signals.
The performance of NeuralEQ with quantized channel signals, for channels of loss -ddB, -12dB, -1ddB, and -21dB, shown in Figure The quantization range set the range of the channel signal without probabilistic noise, given the channel loss. The quantization bit number was set between 5 to bits. Other training and evaluation parameters unchanged. The performance of pruned NeuralEQ without quantization is also shown for comparison.
E COMPARING COMPLEXITY OF NEURALEQ WITH FC NEURAL NETWORK
In this section, evaluate how the proposed NeuralEQ efficiently mimics the optimum decoder, comparing it with the fully connected network. The fully connected neural network can also mimic arbitrary function, but the problem is that the complexity is too high. By quantifying the dif- ference in complexity to achieve target performance, we can claim the efficiency of the proposed NeuralEQ.
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bevt Wendioo
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Figure 9: Sparsity of each NeuralEQ layer with thirteen pruning iterations. (a) Pruning for -7dB channel, (b) -12dB channel, (c) -1dB channel, (d) -21dB channel. 10% of total weight is removed per iteration and corresponding BER is written for each iteration.
The proposed NeuralEQ has hyperparameter N, which is the number of features per layer. Choos- ing an optimum N is indeed an important process to go through. There are many ongoing studies about optimizing hyperparameter of the network and researchers have provided different types of search strategies depending on network structure. We choose Tree-Parzan Estimator (TPE) because it is appropriate algorithm for model which has low dimensional search space and NeuralEQ has single hyperparmeter N.
As result, Thoos that the proposed NeuralBQ outperforms FC network for all four channels conditions. NeuralFQ not only provides better BER than FC network does, but also implements fewer parameters than FC network does to run the network.
FC Network
NeuralEQ
SBR (dB)
Structure
BER (e-3)
Parameters
Structure
BER (e-3)
Parameters
7
216, 376
1.126
85,908
N: 22
0.961
18,594 (21.6%)
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[ 408,, 488
2.845
206,852
N:
2.309
15,464 (7.47%)
16
[ [472, 344
12.56
170,228
N: 35
11.221
45,959 (27.0%)
21
(352, 512]
56.054
187,364
N: 29
50.516
31.817 (16.9%)
Table 4: TPE Strategy Results
F SIMULATION RESULTS FOR ADDITIONAL CHANNELS
In addition to the channels used earlier, manually modified channels are used further evaluate the performance of the proposed NeuralEQ. The more diverse channels are shown in Figure TIa Compared to Figure 7b, the number of pre-cursors is increased, and the polarity of random cursors
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are flipped. The length of ISI is large in the order of Custom CH3, Custom CH2, Custom CHI. The simulation performed in Figure 4 is performed again for the new channels. The results are shown in Figure II It confirmed that the proposed NeuralEQ has better BER performance than conventional EQ for the newly generated channels too.
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Figure 11: Performance for customized channel which has more pre-cursor and negative ISI. (a) Single bit response of each customized channel. BER performance for (b) Custom Ch1, (c) Custom Ch2, and (d) Custom Ch3.
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Reviews:
[{"review_id": "lvSBv41iAWp", "reviewer": "Reviewer_X5rc", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper considers the problem of channel equalization for wired-line communications systems,\nand proposes a machine learning based equalizer inspired by forward-backward algorithm.", "summary_of_the_review": "A certain contribution can be found in the proposed method, while a classical toy model is assumed. However, authors also claims the effectiveness of the proposed method in modern wire-line communications systems, which I cannot agree.", "strength_and_weaknesses": "The proposed approach itself might have some novelty, however, I cannot recommend the acceptance \nof the paper, mainly due to the lack of effectiveness.\nSome detailed comments are listed below.\n\n-From the view point of the development of equalization systems, this kind of serial equalizer\nis considered as rather classical approach, and block transmission based equalization scheme\nsuch as OFDM will be the main stream of the equalization scheme nowadays. Thus, more concrete \ntarget of the wired line communications systems including specific standard should be clarified\nin the paper instead of only \"wired-line communication. Especially, it is hard to imagine a\nwired-communications systems which have difficulty to use PAM4 in this era.\n\n-The complexity of the proposed approach is compared as that of the FB algorithm to show the \neffectiveness of the proposed approach, but it should be compared with frequency domain equalization\napproach, which has much lower computational complexity of O(N log N).\n\n-In the performance evaluation, the channel models used in the simulations will not be sufficient.\nIt is true that a certain pre-cursors can be recognized, but judging from Fig. 7, all channel models\nare almost minimum phase channel. Please show the performance in much more non-minimum phase channel\nas well.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "RI8FBgxZt6W", "reviewer": "Reviewer_Qiy5", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "UPDATE: I have read the rebuttal and updated the score.\n\nXXXXXXXXXXXXXXXXXXX\n\n\nThis paper addresses the problem of extremely fast wired connections, such as in very fast ethernet or PCIe connections. The main issue when increasing the communication speed is ISI (inter-symbol interference), which is traditionally solved using an equalizer. A very good equalizer is the forward-backward algorithm, but it is prohibitively high in computational complexity. This paper presents a learned equalizer, which has lower computational complexity in some cases, while retaining good bit error rate performance.\n", "summary_of_the_review": "Relatively good work with some flaws that would warrant a new revision.\n", "strength_and_weaknesses": "Strengths:\n\n—-------------------\n\nThe paper has many strengths and is tackling an important problem. For the sake of time  I am only writing about weaknesses in this review, since those are the ones that should be actioned upon.\n\nWeaknesses (W)\n\n—--------------------\n\nW: this paper is about applying neural network to replace a specific algorithm, namely the equalizer of a wired connection receiver. This makes the impact of the paper low for the machine learning community. Which of the finding can be generalized to other domains and how? If this cannot be described, maybe this paper is better suited in the publications in the communications domain.\n\nW: The authors do not extensively study the reasons behind the good performance of the NeuralEQ nor they discuss how the resulting NeuralEQ is different from the Forward-Backward algorithm, from which it is motivated. It is clearly different since it has lower complexity in many cases. Is it because of some assumptions in the Forward-Backward algorithm are wrong?\n\nW: The authors do not discuss the training and validation data split in detail. The training and validation data should be described in detail in every setting. \n\nW: How general is the trained algorithm? Does the algorithm need to be retrained if something in the transmission path changes? What are the typical changes? How often in real life would this happen?\n\nW: How representative are the simulations to the real world? Since all results are produced by simulations, the authors should cite previous works that have compared similar simulations to real channels. The best option would of course be to record real data and validate the simulation trained algorithm on real data.\n\n\nMinor weaknesses and comments:\n\n—---------------------\n\nAbstract: “Rapid growth of ML applications demands high-performance computing systems\nto perform massive data processing.”, I don’t think ML applications are the only ones that require high bandwith, e.g., video applications is pretty large.\n\nMinor spelling mistakes:\n\nHeader 4: “EXPERMIENT RESULTS” -> EXPERIMENT RESULTS \n\n4.2 “NerualEQ” -> NeuralEQ \n\nFigure 4: “pruning increase\nthe performance of NeuralEQ.” 🡪 … pruning increases the performance of NeuralEQ\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "1D-1MV1MBl4", "reviewer": "Reviewer_HhyM", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a neural network-based equalizer that mimics the forward-backward algorithm for wireline communication. The authors also tried pruning the neural network.", "summary_of_the_review": "Overall, the proposed idea is interesting. It has lower complexity than the optimal FB equalizer while achieving better BER performance than conventional equalizers. However, the authors did not provide a quantitative comparison for the computational complexity. How much more computation cost does the FB algorithm have compared to the proposed solution? Also, it is unclear whether the computation cost of the proposed solution is much higher than conventional equalizers.", "strength_and_weaknesses": "Strength:\nThe proposed solution has lower computational complexity than the forward-backward (FB) algorithm, while the BER performance is better than conventional equalizers.\n\nWeakness:\n1. The BER performance of the proposed solution is worse than the FB algorithm, especially in the high SNR region. \n2. The authors did not provide a quantitative comparison between the proposed solution and the FB algorithm. The authors did not provide any detailed FLOPs numbers.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject