**Nonlinear Distortion Correction in Comms Channel Paths: Conventional techniques may not be sufficient, but a new approach to implementing nonlinear algorithms may accomplish the goal.**

By Roy Batruni, CEO, Optichron

Nonlinear signal distortion is widespread and complex. Signal distortion is a pervasive impairment in communications and mixed-signal environments. Signals that propagate through a medium are degraded by changes in their amplitude, phase and frequency properties.

The first two effects are caused by linear distortion and the latter by nonlinear distortion. Most real-life applications are dominated by linear distortion, which falls within the realm of linear system theory and linear signal processing-a mature science that has benefited from decades of advancements in theory, algorithms and applications.

Nonlinear distortion, on the other hand, is a much more difficult problem that has not been addressed with a comprehensive, unified set of advancements. Consequently, a robust approach to modeling and countering nonlinear impairments still does not exist.

Spectral growth, as well as in-band signal corruption, degrades the signal-to-noise-and-distortion ratio, which consequently lowers the channel data capacity. At the physical level, nonlinear distortion is caused by channel or circuit elements whose values change as a function of the signal levels. This in turn changes the channel parameters, and a snowball effect ensues.

This effect is absent in linear systems, and therefore linear signals can only be modeled as multidimensional planes in signal space. Because of the dependence of signals on changes in channel parameter values, however, nonlinear signals can only be modeled as a curved manifold in the signal space. Therefore, an infinite number of possible shapes to the nonlinear transfer function exist. As a consequence, there is no single unifying nonlinear system theory that can model the dynamics, stability, invertibility or time and frequency properties of a nonlinear system. An equivalent form of linear system theory in the nonlinear domain remains elusive.

Conventional nonlinear techniques have drawbacks **(References 1 and 2)**. Nonlinear signal-processing techniques have taken diverse and varied paths. One class of solutions focuses on Volterra and Weiner approaches. A Volterra model is used to decompose any nonlinear function into an infinite series whose variable terms are constituted of higher powers of the signal and its past values, as well as cross products of those terms. Fixed coefficients of the series determine the actual properties of the nonlinear model. A Weiner model is similar; it combines linear distortion effects along with the nonlinear Volterra model.

Several drawbacks to such techniques have limited their utility. Signal term cross products and higher powers result in very high implementation complexity. Gaussian noise properties of the signal change dramatically at the Volterra filter output and cause large errors in communications applications. Inverting a Volterra series of even a few terms requires a highly complex infinite series model. Last, adapting a Volterra or Weiner filter is almost impossible because the variables are lumped into cross products. This means that signal terms contribute to updating more than one coefficient simultaneously, thus generating local minima in the adaptation-error surface.

**A new approach to nonlinear filters**

Optichron’s engineering team approached the problem of designing nonlinear filters at the root, with state-space models that deviate from linear system theory, in that their state variables change as a function of a varying, rather than a fixed, state-transition matrix. The varying state-transition matrix is a function of past signals, state variables and past values of the matrix itself. The actual filters that are implemented in practice are directly synthesized from the new state-space models. The resulting filters appear to be "linear" in that the input signal is convolved with the filter coefficients, but the filters are nonlinear because the filter coefficients are not fixed, but vary as a function of state-space parameters. Thus, the nonlinear coefficients "modulate" the filter input signal to generate a nonlinear output signal. This is a direct reflection of the physical nonlinear effects.

There are many ways to structure the implementation of the filters. The simplest approach is to use a bank of simplified nonlinear filters whose coefficients vary as a simple function of past inputs, outputs and state variables. Their outputs are summed together to form a more powerful composite nonlinear filter whose coefficients are a highly complex function of past inputs, outputs and state variables.

The new approach has several theoretical and practical benefits. At each instant in time, the filters perform a linear function. In this new construct, filters and their inverse have the same complexity or order-something not possible with nonlinear functions using conventional approaches. Furthermore, robust stability criteria that allow for designing inverse filters mean that filters with poles, zeros or both are straightforward to implement. Consequently, finite and infinite impulse response nonlinear filter designs are easy to generate. The modular and simple structure of the filters allows for simple scalability between applications with small or large amounts of nonlinear distortion. Higher-order filter banks can address applications with much more severe nonlinear impairments than low-order banks. Last, the simple structure of the filters allows for architecting adaptive nonlinear filters that can track changes in the nonlinear parameters of a distorted channel or circuit.

**A practical implementation**

The set of comprehensive nonlinear signal-processing theorems, algorithms and filtering techniques that was developed by Optichron has been distilled into a powerful signal-processing engine architecture that the company refers to as Turbolinear technology. Figure 1 shows a block diagram of the architecture in which the composite engine is formed from a nonlinear filter bank whose outputs are summed together.

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Figure 1: Linearizer IC block diagram with embedded Turbolinear architecture, with linearized buffer amplifier |

This engine is embedded in the company’s Linearizer IC, which targets nonlinear distortion in data-conversion applications. The Linearizer IC uses a bank of 16 simple, nonlinear filters whose coefficients are a function of signal history and state-space parameters. The outputs of the filters are summed together to form a powerful composite filter that can counter any type of nonlinear distortion: static, dynamic, continuous or discontinuous. The interpolator block re-generates signals that are subsampled from higher Nyquist zones by the analog-to-digital converter (ADC). This allows the filter bank to eliminate the harmonics that are aliased into the baseband.

Buffer amplifier and ADC nonideal circuit behavior causes the signal path through both components to be nonlinear. Figure 2a shows a spectral plot of a commercial 14-bit 105-megasample per second (MSPS) amp/ADC combination, where the fundamental signal tone is digitized along with a multitude of undesired harmonics caused by nonlinear distortion.

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Figure 2a: Spectrum of digitized sinusoid using a commercial 14-bit, 105-Msps ADC and buffer amplifier. Measured SFDR is 66 dB. |

When this distorted signal is processed by the Tubolinear engine, the nonlinear filters invert the effects of the nonlinear amp/ADC signal path. Figure 2b shows the spectral plot of the linearized signal, where the harmonics arising from nonlinear distortion have been eliminated.

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Figure 2b: Spectrum of same digitized sinusoid after processing with Optichron’s Linearizer IC. Measured SFDR is 92 dB. |

Figures 3a and 3b illustrate the performance of the Turbolinear engine across the entire fourth Nyquist zone, where the input frequency sweep range covers 150 to 200 MHz.

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Figure 3a: Performance in the 4th Nyquist zone: SFDR and THD measurements of the ADC output before processing with Optichron’s Linearizer IC. |

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Figure 3b: Performance in the 4th Nyquist zone: SFDR and THD measurements of the ADC output after processing with Optichron’s Linearizer IC. |

Post-processing the ADC output signal with this technology clearly improves the signal conversion spurious-free dynamic range (SFDR) by 24 dB and allows for wider signals at higher frequencies to be subsampled. This feature is increasingly important in communications, medical imaging, instrumentation, test and measurement, and military applications.

An illustration of the actual effect of nonlinear distortion in communications channels is shown in Figure 4a.

Here, a 128-QAM constellation is transmitted through an amplifier with amplitude-amplitude and amplitude-phase nonlinear distortion, and the plot shows the received constellation after equalization with a theoretically optimal linear equalizer. The remaining degradation is due to nonlinear distortion, as can be seen in Figure 4b.

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Figure 4b: Constellation of the same digitized sinusoid after processing with Optichron’s Linearizer IC. Measured SFDR is 92 dB |

where a nonlinear equalizer using the Turbolinear architecture is applied to the signal in Figure 3a, and the constellation points are restored to a high degree of separation.

**References**

1. "Identification and Compensation of Nonlinear Distortion," John Tsimbinos, PhD. Dissertation, Institute for Telecommunications Research, School of Electronic Engineering, University of South Australia, The Levels, Australia. February 1995.

2. Nonlinear System Theory: The Voltera/Weiner Approach, Wilson J. Rugh, Johns Hopkins University Press, 1981 (ISBN O-8018-2549-0).

**About the author**

Roy Batruni has 25 years of IC industry experience as a technologist and executive, specializing in communications IC architecture and design, with a deep emphasis on signal processing. His experience includes National Semiconductor’s Telecom, Mass Storage and DSP groups; ControNet; Enable Semiconductor; and Avio Digital. He founded Optichron in 2003.