Novel low resolution ADC-DSP optimization based on non-uniform quantization and MLSE for data centers interconnects

Yaron Yoffe; Dan Sadot

doi:10.1364/OE.24.005346

1. Introduction

Over the last few years, bandwidth requirement of data centers optical interconnection has grown tremendously [1]. Accelerated global internet usage, as well as newly emerging cloud computing, big data and social media applications results in massive capacity requirement of both intra-connections between switches (e.g., top of the rack switches) inside the data-centers element, as well as external inter-connections between different data-centers locations. In turn, the above optical connections are expected to scale up in speed, and support ultra-high data rates, i.e. 100Gbit/sec and beyond, under severe constrains of power consumption and cost [2]. In particular, the proposed solutions should be based on existing infrastructure, which includes low-costs and severely bandlimited opto-electronic components such as digital-to-analog-converters (DACs), electrical drivers, modulators, optical receivers and ADCs.

Among emerging solutions is the incorporation of DSP dedicated integrated circuits (IC), which can effectively compensate for the optical fiber and opto-electronic devices impairments [3–5]. Such system architecture provides a cost effective solution, as it leverages on Moore’s ‘law’ for low-cost, low-power consumption and high speed implementation, instead of introducing costly wide-bandwidth opto-electronic components. Specifically, it was demonstrated [4] that maximum-likelihood sequence estimation can effectively mitigate the inter-symbol-interference (ISI) resulting from system bandwidth limitation, as well as for different impairments of the optical fiber. However, one major bottlenecks in such design is the high-speed and high-resolution ADC device that should meet the system demands [6]. As the analog bandwidth increases, the complexity of high-resolution ADCs circuitry is scaling drastically, resulting in excessive power consumption and extremely high costs [7]. In addition, the high resolution ADCs require matching high resolution DSP units, which also increases the system complexity.

One way to overcome this technological challenge is to reduce the ADCs physical number of bits (PNOB) and thus significantly decrease the amount of electronic hardware (and accordingly the power dissipation and cost). The tradeoff in this case is introducing large quantization distortions, which may seriously degrade the bit-error-rate (BER) performance.

Typically the ADCs are based on uniform quantization, i.e. the threshold and output levels are uniformly distributed within the signal dynamic range. When the quantization is sufficiently fine, the quantization distortion can be modeled as uniformly distributed white noise [8,9], and incorporated into the BER analysis [10–12]. However, in case of low-resolution quantization (1-4 bits), the distortion is becoming severe, non-linear, and deviates from the additive noise model. In turn, the effect of BER performance is less predictable, and depends strongly on the calibration of the quantization thresholds and output values.

In this paper, a novel low-resolution and non-uniform quantization method is proposed, optimized for MLSE based receivers. The non-uniform quantization allows better utilization of the quantization bits, as the threshold levels are determined based on the signal statistics. Differ from conventional non-uniform quantizers such as the Lloyd-Max (LM) quantizer [13,14], here the optimization is based on detection criterion (BER) instead of the mean-squared-error (MSE) metric. In order to incorporate the precise effect of ADC thresholds calibration on the BER performance, an inclusive distortion model is introduced which considers the combined effect of the channel deterministic impairments, additive noises, and ADC quantization, similarly to [15]. This model, denoted in this paper as quantized noise model (QN), is based on transition probabilities from the channel deterministic branches into the discrete ADC quantization regions (or bins), rather than on arithmetic errors that are based on real-valued ADC outputs. This enables the optimization of a non-uniform quantization scheme, such that maximal statistical separation between different paths on the MLSE receiver trellis diagram is achieved.

Although the concept of BER optimal ADC was already reported in [16], here the optimization is extended for MLSE detection, taking advantage of its inherent robustness to non-linear distortions, and its ability to compensate for signal-dependent patterns [5]. In summary, the quantization is treated as a highly non-linear impairment, and the thresholds are optimized inclusively with the MLSE detector to allow sequence detection with minimal errors.

Through a set of Monte-Carlo simulations it is demonstrated that the proposed method leads to substantial signal-to-noise-ratio gain, and thus mitigates the penalty usually caused by quantization distortions. It is also shown that the MLSE-targeted quantization improves significantly the performance as compared to the MSE-based optimal LM quantizer in term of BER.

The reset of the paper is organized as follows. In section 2 the quantization distortion is modeled along with its mathematical conditions (for its non-linear model versus the additive noise model). In section 3, the alternative quantized channel model is presented, and is used to thoroughly analyze the BER performance, while considering the exact calibration of the quantization thresholds. This result is used in section 4 to optimize the quantization such that the effect on the BER is minimized. The last sections include Monte Carlo simulation results followed by discussions and conclusions.

2. Quantization – additive white noise versus non-linear distortion

The purpose of quantization is mapping continuous analog signals into a finite set of values. The ADC dynamic range is divided into regions ${r_{i}}_{i = 1}^{K}$ , which are separated by analog thresholds ${t_{i}}_{i = 1}^{K + 1}$ , and are represented with real-valued numbers (ADC outputs). The number of regions is determined by the number of quantization bits $B$ and is given by $K = 2^{B}$ .

Without loss of generality, the quantization distortion can be modeled as the difference between the sampled ADC input and the corresponding quantized value:

q (n) = y (n) - D {y (n)}

where

y (n)

is the received sampled signal,

D {\cdot}

refers to the quantization function, and

D {y (n)}

is the corresponding ADC output value.

In ADC based digital receivers, it is important to understand the fundamental properties of the quantization distortion and its effect on the system performance. This topic is widely described in the literature, with special emphasis on uniform quantizers [8,9]. The detailed analysis, provided in [8, chap.5] demonstrates that under certain condition, the distortion can be characterized as an additive noise, uniformly distributed and uncorrelated with the input. This condition is mathematically well established, and depend on the quantization step size $Δ$ which is the analog distance between the thresholds, i.e., the size of each quantization bin. Conversely, the condition can be manifested in relation to the bandwidth of the characteristic function $Φ_{x} (u)$ , which is the Fourier transform of the analog signal probability density function (PDF). A sufficient condition is given in [8,9]:

Φ_{x} (u) = 0 f o r | u | > \frac{2 π}{Δ}

Once this condition is met, and the additive noise model is valid, the quantization noise variance is approximately given by $σ_{q}^{2} = \frac{Δ^{2}}{12}$ and the signal-to-quantization-noise ratio (SQNR) can be approximated as follows [7–9]:

S Q N R = 6.02 B + 4.8 - 20 \log_{10} \frac{R}{2 σ_{y}}

where

R

is the input signal dynamic range and

σ_{y}

is its variance. In this case, the degradation in BER performance can be analytically computed in intensity modulation signals [10], complex constellations [11] and in linearly equalized channels [12].

Nevertheless, here we focus on low-resolution ADCs and non-uniform quantizers, in which the condition above is not satisfied. Consequently, the quantization effect generates signal dependent patterns, as there exist a correlation between the input and the error signal of Eq. (1). This indicates that the distortion cannot be analyzed in terms of random independent noises, rather, it should be treated as a deterministic, non-linear impairment. Subsequently, the MLSE decoder can be applied to mitigate the non-linear effect on the BER performance.

3. Error probability of MLSE receiver in the presence of low resolution and non-uniform quantization

In this section, a thorough analysis of BER performance in the presence of low-resolution and non-uniform quantization, followed by MLSE equalizer is presented. In order to incorporate the effect of the quantization non-linearity, explained above, an alternative statistical quantized noise channel model (QN) is adopted. In turn, in the following section inclusive optimization is performed in order to minimize the BER.

The first step of the analysis is observing the well-known general expression which provides a tight upper bound and a close approximation for the error probability in MLSE based detectors [5,17]:

P_{e r} \leq \sum_{ε} W_{H} (ε) P (ε) = \sum_{ε} W_{H} (s_{i} \neq s_{j}) P (s_{i} | s_{j}) P (s_{j})

where

ε

refers to an error event,

W_{H} (ε)

is the corresponding Hamming weight, or the number of symbol errors.

P (s_{i} | s_{j})

is the probability of a error event, where

s_{j}

and

s_{i}

are the transmitted and decoded sequences respectively, and

P (s_{j})

is the a-priory probability that the sequence

s_{j}

was transmitted. Assuming a channel with ISI and AWGN, each ADC sample can be modeled as a sum of a deterministic analog branch

x (n)

and additive noise

z (n)

:

y (n) = x (n) + z (n)

Consequently the term

P (s_{i} | s_{j})

in Eq. (4) is defined by the Euclidean norm associated with the error event

ε

:

P (s_{i} | s_{j}) = Q (\frac{‖ x_{i} - x_{j} ‖}{2 σ_{z}})

where

x_{i} = (x_{1}, x_{2} ... x_{N}) \in R^{N}

and

x_{j} = (x_{1}, x_{2} ... x_{N}) \in R^{N}

represent vectors of

N

consecutives deterministic analog branches (noiseless) that correspond to the sequences

s_{i}

and

s_{j}

respectively.

σ_{z}

is the AWGN standard deviation and

Q (\cdot)

represents the Gaussian Q function.

Here, the analysis is focused on the incorporation of the effect of severe and non-uniform quantization. In this approach the quantization function is combined with the channel additive noises and other random impairments into an inclusive distortion model, similarly to the approach used for the mutual information derivations in [15]. This generalized QN model, $D {x + z} = D {y}$ , operates on the deterministic analog channel branch $x$ , and is defined in terms of transition probabilities from the analog branches into the quantized values, rather than in terms of arithmetic errors (the Euclidean norms in Eq. (6). Since a large error is expected between the analog received signal $y (n)$ and the quantized value $D {y (n)}$ , the latter will be assigned to an analog region $r$ instead of the real-valued number of the ADC output by the ADC-DSP block as shown in Fig. 1. The ADC-DSP block has a two-fold operation: (1) generation of the conditional PDFs engines for the MLSE, and (2) computation of the transition probabilities from $x$ to $r$ , as in Eq. (7). In step one, it should be emphasized that the conditional PDFs engines are generated through a non-parametric histogram estimation method [3,5], rather than using the parametric method of moments approach which is based on Gaussian distribution [5]. The computations are based on the probability of $y (n) = x (n) + z (n)$ falling within a specific region $r_{i}$ (quantization bin), and is derived by the probability mass of $y (n)$ accumulated in each of the regions:

\begin{array}{l} P {D {y} = r_{i} | x} = P {D {x + z} = r_{i} | x} = p_{r} (x + z \in r_{i}) = p_{r} (t_{i} < x + z \leq t_{i + 1}) \\ D {y} = D {x + z} = {\begin{matrix} r_{1} \\ r_{2} \\ ⋮ \\ r_{K} \end{matrix} \begin{matrix} p_{r} (t_{1} < x + z \leq t_{2}) \\ p_{r} (t_{2} < x + z \leq t_{3}) \\ ⋮ \\ p_{r} (t_{K} < x + z \leq t_{K + 1}) \end{matrix}} \end{array}

where

{r_{i}}_{i = 1}^{K}

are the analog regions, and

{t_{i}}_{i = 1}^{K + 1}

(t_{1} = - \infty, t_{k + 1} = \infty)

are the appropriate thresholds between the regions. Therefore, even though two sampling cases may have the same real-value, the proposed algorithm interprets them differently according to the different probability mass of each of the sampled cases (in the same region). In the case where

z (n)

is AWGN, the probabilities can be computed as follows:

p_{r} (y \in r_{i} | x) = \frac{1}{\sqrt{2 π σ_{z}^{2}}} \int_{t_{i}}^{t_{i + 1}} \exp (\frac{{(y - x)}^{2}}{2 σ_{z}^{2}}) d y = Q (\frac{x - t_{i}}{σ_{z}}) - Q (\frac{x - t_{i + 1}}{σ_{z}})

This is illustrated in Fig. 2(a). From (7) and (8) we notice the relation between the transition probabilities and the calibrated ADC thresholds which, in turn, are optimized by the ADC-DCP block. This optimization scheme is derived in the following section.

Fig. 1 Combined ADC-DSP system model. The QN $D {x (n) + z (n)}$ model is operating on the channel deterministic analog branches $x (n)$ . The ADC outputs $r (n)$ , which represent analog regions, are post-processed by the MLSE for data detection. The ADC-DSP block computes (a) the MLSE metrics and (b) the transition probabilities from $x_{i}$ to $r_{i}$ , for each of the channel branches.

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Fig. 2 (a) Each of the channel branches $x_{n}$ can be assigned to one of the four digital output values (assuming 2 bits quantization). The probability to receive each value depends on the channel noises and the analog value of the branch compared to the thresholds (reference levels ${t_{i}}_{i = 1}^{4}$ ). (b) Quantization of a sequence of 2 consecutive ADC samples: $x = (x_{n}, x_{n + 1})$ .

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Follows, if a sequence of $N$ consecutives values $y = (y_{1}, y_{2} ... y_{N})$ is quantized into one of the $K$ regions, there would be $K^{N}$ different combinations. The quantization of a sequence of 2 consecutive samples is illustrated in Fig. 2(b). The probability of an error event $P (s_{i} | s_{j})$ can be determined as follows: the symbols sequences $s_{j}$ and $s_{i}$ are each assigned with $K^{N}$ conditional probabilities (conditional to the transmission of $s_{i}$ and $s_{j}$ , respectively) for each possible combination. The decision between two sequences is made according to ML criterion, i.e. the MLSE decoder decodes each of the combinations into one of the sequences according to the highest conditional probability. An error evet occurs if the received samples of the sequence $s_{j}$ falls into a combination of regions that are assigned with the sequence $s_{i}$ . Note that although the MLSE decision criterion described above is conventional [4,5], the transformation from continuous noise representation into the inclusive QN model allows to form a methodology that depends on the regions (determined by the thresholds) and not on the arithmetic values. Consequently the combination of the QN model and MLSE enables the optimization of the quantization thresholds to achieve minimum BER.

For simplicity and further explanations, the following notations are suggested: $x^{(j)} = (x_{^{1}}^{(j)}, x_{^{2}}^{(j)} ... x_{^{N}}^{(j)})$ represents the analog deterministic channel branches, associated with the sequence $s_{j}$ . $y^{(j)} = (y_{^{1}}^{(j)}, y_{^{2}}^{(j)} ... y_{^{N}}^{(j)})$ are the corresponding analog received samples (including the channel noises and other random physical impairments such as clock jitter and fiber impairments). $D {y^{(j)}}$ refers to the digital quantized values (or regions). All the combinations of regions that are decoded into $s_{j}$ are denoted as $R^{(j)} = (r_{^{1}}^{(j)}, r_{^{2}}^{(j)} ... r_{^{N}}^{(j)})$ , and all the combinations of regions that are decoded into $s_{i}$ are denoted as $R^{(i)} = (r_{^{1}}^{(i)}, r_{^{2}}^{(i)} ... r_{^{N}}^{(i)})$ such that $| R^{(i)} | + | R^{(j)} | = K^{N}$ . The probability of an error event from $s_{j}$ to $s_{i}$ is a sum of probabilities such that $D {y^{(j)}} \in R^{(i)}$ :

P (s_{i} | s_{j}) = \sum_{R^{(i)}} P (D {y^{(j)}} \in R^{(i)}) = \sum_{R^{(i)}} P (D {y_{1}^{(j)}} = r_{1}^{(i)} \cap D {y_{2}^{(j)}} = r_{2}^{(i)} ... \cap D {y_{N}^{(j)}} = r_{N}^{(i)})

Assuming independent noises, Eq. (9) can be rewritten as:

P (s_{i} | s_{j}) = \sum_{R^{(i)}} P (D {y_{1}^{(j)}} = r_{1}^{(i)}) \cdot P (D {y_{2}^{(j)}} = r_{2}^{(i)}) ... \cdot P (D {y_{N}^{(j)}} = r_{N}^{(i)})

Substitute Eq. (8) into Eq. (10) and Eq. (10) into Eq. (4) yields a closed form approximation for the BER upper bound in case of low-resolution or non-uniform quantization in channels with memory and AWGN:

\begin{array}{l} P_{e r} \leq \sum_{ε} W_{H} (ε) P (ε) \\ = \sum_{ε} W_{H} (s_{i} \neq s_{j}) {\sum_{R^{(i)}} P (D {y_{1}^{(j)}} = r_{1}^{(i)}) \cdot P (D {y_{2}^{(j)}} = r_{2}^{(i)}) ... \cdot P (D {y_{N}^{(j)}} = r_{N}^{(i)})} P (s_{j}) \end{array}

However, this computation in practical application may become tedious due to the large number of possible error events. Instead, and similarly to the non-quantized case, the computation may be simplified by focusing on the dominant term in the summation [17]. The exponential dependence of each term in the sum causes the expression to be dominated by one of the error events, and the error probability can be approximated as follows:

\begin{array}{l} P_{e r} \leq \sum_{ε} W_{H} (ε) P (ε) \approx W_{H} (ε_{\max}^{j - > i}) P (ε_{\max}^{j - > i}) \\ \approx W_{H} (s_{i} \neq s_{j}) {\sum_{R^{(i)}} P (D {y_{1}^{(j)}} = r_{1}^{(i)}) \cdot P (D {y_{2}^{(j)}} = r_{2}^{(i)}) ... \cdot P (D {y_{N}^{(j)}} = r_{N}^{(i)})} P (s_{j}) \end{array}

Where

ε_{\max}^{j - > i}

is the error event with the highest probability (error from a sequence

s_{j}

into the sequence

s_{i}

).

4. Optimizing the quantization for minimum BER

In the previous section, an inclusive theoretical analysis of the BER at the output of the receiver based on low resolution non-uniform quantization and MLSE detector was derived. This BER expression can be used as an ultimate optimization criterion which allows analytical computation of the thresholds levels that result in minimum BER. Furthermore, this optimization does not involve the quantization output real-valued number, a fact that significantly reduces the computational complexity.

Since the cost function is highly non-linear and has large number of discontinuities (when Eq. (12) is calculated over different error events) we obtain the optimization by a two-step process of pattern search [18] and iterative local optimizations. The first step may be regarded as the acquisition part which includes a derivative free pattern search over all possible thresholds combination (assuming low-resolution, the computation complexity is feasible). The acquisition is made as a first step in order to narrow down the possible range of values of each of the thresholds, and in order to identify the error event of highest probability. Once selected, the second step takes place, which may be regarded as the tracking part. It is performed locally, using the gradient decent algorithm, applied to Eq. (12), to iteratively determine the optimal thresholds.

5. Results

Simulation methodology

An inclusive set of Monte-Carlo simulations was performed, evaluating the performance of the proposed non-uniform quantization method. The model is based on typical datacenters interconnection systems, and includes severely bandlimited opto-electronic and electrical components, and small amount of residual dispersion accumulated over 1km transmission. The simulation was performed for NRZ OOK modulation, based on intensity modulation and direct detection transmission (IM-DD). It was assumed that the transmitter operates at 56Gbauds, whereas the overall 3-dB analog bandwidth is approximately 20GHz. The resulting ISI and channel impulse response (CIR) is depicted in Fig. 3. Electrical SNR was synthesized by injecting AWGN at the receive side, immediately after the optical-to-electrical conversion (OEC). The ADC building block consists of a track-and-hold (T&H) circuit and an amplitude quantizer, with varying threshold levels. The ADC is followed by an MLSE based detector, containing 8 states and 16 branches, which was implemented by means of the histogram estimation method [5].

Fig. 3 Channel impulse response.

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Simulation results were computed for different number of quantization bits in order to demonstrate the degradation in performance when the ADC resolution is decreased. The results are presented for uniform quantizers and compared with the infinite precision ADC (without quantization distortions). In Fig. 4. it is shown that a 4 bits uniform quantization yields relatively small SNR penalty at BER of 1e-3, which is often considered a threshold for implementing forward-error-correction (FEC) codes. It also supports KP4 FEC, which is the selected FEC scheme for 400G datacenters connections [19] that requires pre-FEC BER value of 2e-4, with SNR penalty of less than 1 dB. The 3 bits uniform quantizer imposes SNR penalty of approximately 2.5 dB at BER of 1e-3 and 4 dB at BER of 2e-4. The emphasis in this paper is on the 2 bits quantizer, in which the condition from Eq. (2) is not satisfied. In this case, the quantization non-linearity is clearly observed by the slope flattening of the BER curve, due to the signal dependent quantization distortion effect. The BER values are higher than 1e-3 at significantly higher SNR values.

Fig. 4 BER vs. SNR curve for an infinite precision ADC, 2 bits uniform ADC, 3 bits uniform ADC and 4 bits uniform ADC respectively.

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Low-resolution non-uniform optimal quantization with MLSE

Here a quantitative analysis is performed for the proposed optimal low resolution non-uniform quantizer with MLSE, which is also compared with the LM quantizer and with “standard” uniform quantizer. Figure 5. summarizes the comparison analysis results. It is shown that while using very low resolution ADC of 2-bits only, the proposed method performs surprisingly well and introduces only 1dB of SNR penalty at BER of 1e-3, as compared to an infinite precision ADC. For comparison, the MSE-based LM quantizer analysis results are presented, indicating that for the case of 2-bits of resolution, a large penalty of 7.5dB is introduces. The result is also compared to the the "standard" uniform quantizer case, indicating that the 2 bits MLSE optimal ADC outperform a uniform quantizer with 3 bits of resolution. In summary, it is shown that a significant improvement can be achieved if the ADC thresholds are optimally calibrated for the MLSE receiver, based on the minimum BER criterion developed here.

Fig. 5 BER vs. SNR curve for an infinite precision ADC, 3 bits uniform ADC, 2 bits uniform ADC, 2 bits Loyd-Max (LM) ADC and a 2 bits MLSE optimal ADC respectively.

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6. Conclusion

In this paper, a novel low-resolution non-uniform quantization method is proposed optimized for MLSE based receivers. By adopting a combined noise and quantization (QN) distortion model, the effect of ADC thresholds calibration on the BER performance was derived. This model enabled the incorporation of the low-resolution and non-uniform quantization non-linearity into the BER computation, such that inclusive optimization could be performed directly on the BER target function. Through Monte Carlo simulations, it was demonstrated that the proposed method substantially improves the BER vs SNR curve, with significant gain over existing methods. The approach can also be applied for multi-level PAM, with higher ADCs resolution or smaller memory depths. This may enable the use of low cost and low power consumption ADC components, which is critical for the development of next generation datacenters connectivity.

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Novel low resolution ADC-DSP optimization based on non-uniform quantization and MLSE for data centers interconnects

Abstract

1. Introduction

2. Quantization – additive white noise versus non-linear distortion

3. Error probability of MLSE receiver in the presence of low resolution and non-uniform quantization

4. Optimizing the quantization for minimum BER

5. Results

Simulation methodology

Low-resolution non-uniform optimal quantization with MLSE

6. Conclusion

References and links

Cited By

Figures (5)

Equations (12)

Optics Express