1. Bibliographic Information

1.1. Title

Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems

1.2. Authors

Dominique N. Godard. At the time of publication, he was with the IBM Centre d'Etudes et Recherches in La Gaude, France.

1.3. Journal/Conference

This paper was published in the IEEE Transactions on Communications. It was previously presented at the Ninth Annual Communication Theory Workshop in April 1979. IEEE Transactions on Communications is one of the most prestigious and high-impact journals in the field of telecommunications and signal processing.

1.4. Publication Year

The paper was officially published on November 1, 1980.

1.5. Abstract

The paper addresses the challenge of adaptive channel equalization in digital communication systems without the need for a known "training sequence." Traditional methods require the receiver to know exactly what data is being sent initially to calibrate the system. Godard proposes a new class of nonconvex cost functions that characterize intersymbol interference (ISI) independently of the carrier phase and the specific symbol constellation used. This allows the equalizer to converge (calibrate itself) without requiring carrier recovery first. Once the equalizer has stabilized, carrier phase tracking can proceed in a "decision-directed" mode. The mathematical analysis of convergence is provided and validated through computer simulations.

1.6. Original Source Link

The original source is available via IEEE Xplore: Original Source Link.

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2. Executive Summary

2.1. Background & Motivation

In digital communications, signals sent over a physical medium (like a phone line or radio airwaves) suffer from distortion. This distortion causes symbols to "smear" into one another, a phenomenon known as intersymbol interference (ISI). To fix this, receivers use an equalizer—a digital filter that attempts to reverse the channel's distortion.

Traditionally, equalizers require a training period: the transmitter sends a specific, pre-arranged sequence of bits that the receiver already knows. The receiver compares what it got with what it expected to get, calculating an error and adjusting its internal settings.

The Problem: In "multipoint networks" (where one central hub talks to many devices), retraining is a major bottleneck. If a new device joins or a connection drops, the hub might have to stop everything to resend a training sequence. This kills "data throughput" (the amount of useful data sent per second).

The Motivation: Godard wanted a way for a receiver to "blindly" equalize the signal—meaning it can figure out how to fix the distortion just by looking at the corrupted signal itself, without needing a known training sequence.

2.2. Main Contributions / Findings

1. The Godard Algorithm (CMA precursor): He introduced a new family of cost functions called dispersion functions. These functions allow the equalizer to minimize ISI without knowing the transmitted symbols and without knowing the phase of the carrier wave.

2. Carrier Independence: He proved that the equalization process could be decoupled from the carrier recovery process. This is a significant leap because, normally, you need a "locked" carrier to even begin to understand the data.

3. Mathematical Convergence Proof: He analyzed the "nonconvex" nature of these functions (meaning they have multiple "dips" or local minima) and provided a strategy for initializing the equalizer to ensure it reaches the correct, global minimum.

4. Practical Robustness: Through simulations, he demonstrated that the algorithm works even under severe distortions and noise for high-speed data rates (up to 12,000 bits/s).

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3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

3.1.1. Quadrature Amplitude Modulation (QAM)

QAM is a method of sending digital data by changing (modulating) the amplitude of two carrier waves that are 90 degrees out of phase (quadrature).

• In-phase (I): The cosine component.
• Quadrature (Q): The sine component. In this paper, these are treated as complex numbers, where the real part is I and the imaginary part is Q. A constellation is a map of all possible (I, Q) points.

3.1.2. Intersymbol Interference (ISI)

Imagine trying to yell a series of words through a long, echoey tunnel. If you yell too fast, the echo of the first word overlaps with the sound of the second word. In electronics, this "echo" is caused by the physical properties of the cable or air. This is ISI.

3.1.3. Equalization

An equalizer is a filter (usually a "tapped delay line") that acts like an "anti-echo" machine. It tries to subtract the overlapping parts of the signal so that each symbol stands clearly on its own.

3.1.4. Carrier Phase Recovery

Signals are sent at high frequencies (the carrier). The receiver must perfectly match the frequency and phase of that carrier to extract the data. If the phase is off by even a few degrees, the (I, Q) points in the constellation will appear "rotated," leading to errors.

3.2. Previous Works

The most relevant prior work mentioned is by Y. Sato (1975). Sato proposed a "self-recovering" (blind) equalization method for simple amplitude-modulated systems. However, Sato's method:

1. Only worked for 1D signals (like binary or multilevel AM).
2. Could not handle the carrier phase rotation inherent in 2D systems (QAM). Godard's paper is the evolution that brings blind equalization to the complex 2D world.

3.3. Technological Evolution

Before this paper, "blind" equalization was mostly a theoretical curiosity or limited to very simple systems. Godard moved the field into the era of high-speed, 2D modems. His work laid the foundation for the Constant Modulus Algorithm (CMA), which is still a standard tool in digital signal processing today.

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4. Methodology

4.1. Principles

The core intuition is that even if we don't know which specific symbol was sent, we know the statistical properties of the constellation. If the signal is distorted by ISI, the "dots" in the constellation will be scattered. Godard's cost function measures how much the magnitude of the received signal deviates from a specific constant value ($R_p$). By minimizing this deviation, the equalizer naturally eliminates the "echoes" (ISI) that cause the magnitude to fluctuate.

4.2. Core Methodology In-depth (Layer by Layer)

4.2.1. Signal Model and Equalizer Structure

The transmitted signal u(t) is modeled as a sum of data symbols $a_n$ multiplied by a baseband signal element $g_0$ and a carrier wave: $u(t) = \mathrm{Re} \sum_{n} a_{n} g_{0}(t - nT) \exp{j 2 \pi f_{0} t}$

• $a_n$: The data symbol (a complex number).

• $g_0$: The baseband pulse shape.

• $T$: The symbol interval (time between symbols).

• $f_0$: The carrier frequency.

  The following figure (Figure 2 from the original paper) illustrates the two-dimensional transmission system:

  该图像是示意图，展示了一个二维传输系统的结构。图中包含信号输入 $a_n$、重采样模块（Re）、信道模块、相位分离器、均衡器、决策电路以及载波追踪模块。每个模块之间通过连接线标示了信号流向，并体现了信号的相位变化。该图为理解自恢复均衡与载波追踪过程提供了可视化支持。

At the receiver, after some processing, the input to the equalizer is y(t): $y(t) = \sum_{n} a_{n} h(t - nT) \exp j \varphi(t) + v(t)$

• h(t): The overall impulse response of the channel.

• $\varphi(t)$: Time-varying phase shift (frequency offset/jitter).

• v(t): Random noise.

  The equalizer output $z_n$ at time $t = nT$ is calculated using a vector of "tap gains" $c_n$ and the input signal vector $y_n$: $z_{n} = y_{n}' c_{n}$

• $y_n$: The vector of signals currently in the equalizer's delay line.

• $c_n$: The adjustable weights (coefficients) of the equalizer.

• $y_n'$: The transpose of the vector.

4.2.2. The New Cost Function: Dispersion

The central innovation is the Dispersion of order $p$, denoted as $\mathcal{D}^{(p)}$: $\mathcal{D}^{(p)} = E ( \vert z_{n} \vert^{p} - R_{p} )^{2}$

• $E$: The expected value (an average over time).

• $\vert z_n \vert$: The magnitude of the equalizer output.

• $p$: A positive integer (usually 1 or 2).

• $R_p$: A constant calculated based on the signal constellation.

  This function is "blind" because it doesn't compare $z_n$ to a known $a_n$. It only looks at the magnitude $\vert z_n \vert$. Because the phase $\exp(j\theta)$ disappears when you take the magnitude (\vert \exp(j\theta) \vert = 1), this cost function is completely independent of carrier phase errors.

4.2.3. Determining the Constant $R_p$

To ensure the equalizer converges to the "perfect" state, Godard defines $R_p$ based on the statistical properties of the transmitted symbols: $R_{p} = \frac{E \mid a_{n} \mid^{2p}}{E \mid a_{n} \mid^{p}}$

• $a_n$: The theoretical symbols in the constellation. For example, if you use a constellation where all points are on a circle with radius 1, then $R_p = 1$.

4.2.4. The Adaptive Algorithm (Steepest Descent)

The equalizer adjusts its coefficients $c$ by following the "gradient" (the direction of steepest descent) of the dispersion function. The general update rule is: $c_{k+1} = c_{k} - \mu_{p} \left[ \frac{\partial \mathcal{D}^{(p)}}{\partial c} \right]_{c = c_{k}}$

• $\mu_p$: The "step-size" (how fast we change the settings).

  Godard simplifies this into a stochastic approximation algorithm that can be run in real-time on a microprocessor: $c_{n+1} = c_{n} - \lambda_{p} y_{n}^{*} z_{n} \mid z_{n} \mid^{p-2} ( \mid z_{n} \mid^{p} - R_{p} )$

• $\lambda_p$: A small positive step-size.

• $y_n^*$: The complex conjugate of the input signal.

  For the most practical case, $p=2$, the algorithm becomes: $c_{n+1} = c_{n} - \lambda_{2} y_{n}^{*} z_{n} ( \mid z_{n} \mid^{2} - R_{2} )$ This is computationally simple: it only requires a few multiplications and subtractions per symbol.

4.2.5. Combined Architecture

The following diagram (Figure 4 from the original paper) shows how the self-recovering equalizer works alongside carrier tracking:

该图像是示意图，展示了自恢复技术的结构。图中包括了均衡器、决策电路和相关公式。输入信号 y(t) 经过均衡器处理后，产生信号 $z_n$，并与相位信息 $exp^{-j\hat{\phi}_n}$ 进行相乘，进而传递至决策电路，最后输出估计符号 $\hat{a}_n$。同时，均衡器通过方程 Eqn. 22 进行反馈调整，进一步提高系统性能。

1. The equalizer adjusts itself using Eq (22) to remove ISI.

2. The output $z_n$ is rotated by an estimated phase $\hat{\varphi}_n$.

3. The result goes to a "decision circuit" which guesses which symbol was sent ($\hat{a}_n$).

4. The carrier tracking loop uses these decisions to fine-tune the phase.

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5. Experimental Setup

5.1. Datasets and Channels

Godard used computer-simulated channels to test the theory.

• Channels: Two specific channels (Line 1 and Line 2) were modeled with varying levels of amplitude and delay distortion. Line 2 was "extremely severe," representing a very poor-quality telephone line.

• Distortion Characteristics: The following charts (Figure 5 from the paper) show the attenuation (amplitude loss) and group delay (time distortion) for these channels compared to a standard "3002 Basic" line:

  该图像是图表，展示了两个频率通道的衰减和群延迟特性。图(a)和图(b)分别表示通道1和通道2的衰减和群延迟曲线，且均标注了3002 Basic线路的比较数据。这些数据有助于理解通道的性能表现及其对信号传输的影响。

5.2. Signal Constellations

He tested four different types of QAM constellations (Figure 6):

1. 8-Phase: Simple phase-shift keying (PSK).

2. 16-Point Rectangular: A standard grid.

3. 16-Point V29: A specific standard used in 9600 bit/s modems.

4. 32-Point Rectangular: Used for even higher speeds.

   该图像是数据符号星座图示，展示了四种不同的符号星座配置：8-phase、16-point rectangular、16-point V29和32-point rectangular。每种配置均以坐标平面上的点表示，展示了不同的数据传输方案中的符号间隔和布局。这些星座图对理解数字通信中的符号分布和相位对比十分重要。

5.3. Evaluation Metrics

5.3.1. Mean-Squared Error (MSE)

• Conceptual Definition: MSE measures the average squared difference between the equalized signal (after phase correction) and the true intended symbol. A lower MSE means the "dots" are closer to their intended locations.
• Mathematical Formula: $\mathcal{E}^{2} = E \mid z_{n} \exp - j \hat{\varphi}_{n} - a_{n} \mid^{2}$
• Symbol Explanation:
  • $z_n$: Equalizer output.
  • $\hat{\varphi}_n$: Estimated carrier phase.
  • $a_n$: Actual transmitted symbol.

5.3.2. Convergence Speed

This is the number of iterations (symbols) required for the MSE to drop below a certain threshold (e.g., -15 dB), at which point the "eye is open" and the modem can reliably read data.

5.4. Baselines

The primary comparison was against the conventional decision-directed gradient algorithm. Godard noted that for the tested channels, the conventional algorithm has a probability of error close to 1 and fails to converge without a training sequence.

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6. Results & Analysis

6.1. Core Results Analysis

• Successful "Blind" Convergence: The algorithm successfully reduced MSE to the point where the "eye" was open, even when starting with no knowledge of the data.
• Robustness to Distortion: Even on "Line 2" (the severe channel), the algorithm converged.
• Speed: Convergence was found to be relatively slow—taking about 10 seconds at 2400 bauds. While slow compared to trained algorithms, Godard argues this is acceptable for background retraining in multipoint networks.
• p=1 vs. p=2: The version where $p=2$ (using the squared magnitude) generally converged faster than the $p=1$ version.

6.2. Convergence Plots

The following are the results from Figure 7 of the original paper, showing the MSE decreasing over time (number of iterations):

该图像是图表，展示了不同信道条件下均方误差（MSE）随时间变化的收敛速度。图中包含四个部分：左上角(a)和右上角(b)分别表示在$p=1$条件下的Line 1和Line 2的收敛表现，左下角(c)和右下角(d)则显示在$p=2$条件下的相应结果。每部分中标注了不同算法的MSE变化趋势。

• Graphs (a) and (b): Show convergence for $p=1$ on Line 1 and Line 2.
• Graphs (c) and (d): Show convergence for $p=2$. Note that the slope is steeper for $p=2$, indicating faster learning.

6.3. Analysis of Local Minima

Because the cost function is "nonconvex," there is a risk the equalizer could get stuck in a "bad" state. Godard's analysis showed that:

1. The absolute minimum corresponds to zero ISI (perfect equalization).
2. To avoid "bad" local minima, the initialization of the equalizer is critical. He suggests setting one "reference tap" to a value significantly larger than the others.
3. Condition (38) provides a mathematical bound for this initialization: $\mid c_{0} \mid^{2} > \frac{E \mid a_{n} \mid^{4}}{2 \mid h_{0} \mid^{2} ( E \mid a_{n} \mid^{2} )^{2}}$ In simple terms: if you start with one strong tap, the algorithm will find the right solution.

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7. Conclusion & Reflections

7.1. Conclusion Summary

Dominique Godard successfully solved the problem of adaptive equalization for 2D modulation systems without a training sequence. By creating a cost function that ignores carrier phase and focuses on the statistical dispersion of symbol magnitudes, he enabled modems to "self-heal" their connections. This was a landmark contribution to the efficiency of multipoint computer networks.

7.2. Limitations & Future Work

• Convergence Speed: The main drawback is the slow speed (~10 seconds). In modern high-speed contexts, this might be too slow for primary startup, though it remains useful for background maintenance.
• Computational Precision: Godard noted that higher orders of $p$ (like $p=3$) are too sensitive to noise and require too much numerical precision for standard microprocessors of that era.
• Non-convexity: While he provided an initialization strategy, the existence of local minima remains a theoretical concern for extremely unusual channel conditions.

7.3. Personal Insights & Critique

• Innovation: The decoupling of carrier recovery from equalization is brilliant. It bypasses a "chicken-and-egg" problem: you can't recover the carrier without equalizing, and you usually can't equalize without the carrier. Godard broke this loop.
• Impact: This paper is the father of the Constant Modulus Algorithm (CMA). If you use a modern digital communication system today (like cable modems or some wireless links), there is a high probability that a descendant of Godard’s math is running inside the chip.
• Critique: The paper is mathematically dense but very rigorous. One potential area for improvement (though difficult in 1980) would have been testing on real-world non-stationary channels where the distortion changes rapidly. However, for the technology of the time, this was a masterpiece of signal processing theory.