[Paper Review] The Bussgang Decomposition of Non-Linear Systems: Basic Theory and MIMO Extensions

Özlem Tuğfe Demir, Emil Björnson|arXiv (Cornell University)|May 4, 2020

Advanced MIMO Systems Optimization12 references17 citations

TL;DR

This paper provides a rigorous theoretical foundation for the Bussgang decomposition in non-linear signal processing, extending it to complex-valued and MIMO systems. It proves that the output of a non-linearity can be exactly decomposed into a linearly scaled input plus uncorrelated distortion, enabling performance bounds in systems with hardware impairments like quantization and non-linear amplifiers.

ABSTRACT

Many of the systems that appear in various signal processing applications are non-linear, for example, due to hardware impairments such as non-linear amplifiers and finite-resolution quantization. The Bussgang decomposition is a popular tool for analyzing the performance of systems that involve such non-linear components. In a nutshell, the decomposition provides an exact probabilistic relationship between the output and the input of a non-linearity: the output is equal to a scaled version of the input plus uncorrelated distortion. The decomposition can either be used to compute exact performance results or lower bounds where the uncorrelated distortion is treated as independent noise. This lecture note explains the basic theory, provides key examples, extends the theory to complex-valued vector signals, and clarifies some potential misconceptions.

Motivation & Objective

• To establish a mathematically rigorous and unified framework for the Bussgang decomposition in non-linear systems, especially for MIMO and complex-valued signals.
• To clarify and correct common misconceptions in the literature regarding the validity and application of the Bussgang decomposition.
• To extend the classical Bussgang theorem to complex Gaussian random variables and MIMO systems with correlated distortion.
• To demonstrate that the distortion term in the decomposition is uncorrelated with the input but not independent or Gaussian, and to quantify its correlation structure in MIMO systems.
• To provide a generalized Bussgang decomposition for non-Gaussian inputs, showing that the distortion remains uncorrelated with the input but may exhibit non-trivial correlation across multiple outputs.

Proposed method

• Derives the complex Bussgang theorem using minimum-mean-square error (MMSE) estimation, decomposing the output signal into a linear component and an uncorrelated error term.
• Uses the property that uncorrelated jointly Gaussian random variables are independent to prove that the estimation error is uncorrelated with the input, enabling exact decomposition.
• Extends the decomposition to MIMO systems by defining a Bussgang gain matrix $\mathbf{B} = \mathbf{C}_{zx}\mathbf{C}_x^{-1}$, ensuring the distortion $\bm{\eta}$ is uncorrelated with the input $\mathbf{x}$.
• Analyzes the correlation structure of the distortion vector $\bm{\eta}$ via its correlation matrix $\mathbf{C}_\eta = \mathbf{C}_z - \mathbf{B}\mathbf{C}_x\mathbf{B}^\mathrm{H}$, showing it is generally non-diagonal when inputs are correlated.
• Demonstrates through simulations that distortion correlation is significant at low ADC resolution (e.g., 4-bit), but diminishes at high resolution, justifying approximations in some cases.
• Proposes a generalized Bussgang decomposition for non-Gaussian inputs using the linear MMSE estimate of $\mathbf{z}$ given $\mathbf{x}$, with $\bm{\eta} = \mathbf{z} - \mathbf{B}\mathbf{x}$ as the uncorrelated error.

Experimental results

Research questions

• RQ1 How can the Bussgang decomposition be rigorously extended from real to complex-valued Gaussian signals in a way that preserves exactness and uncorrelatedness of the distortion term?
• RQ2 What is the structure of the distortion correlation matrix in MIMO systems, and when can it be safely approximated as diagonal?
• RQ3 How does the Bussgang decomposition perform in non-Gaussian input scenarios, and what are the implications for system performance analysis?
• RQ4 Under what conditions can the uncorrelated distortion in the Bussgang decomposition be treated as independent Gaussian noise without introducing significant error?
• RQ5 Why is the common practice of neglecting distortion correlation in MIMO systems potentially problematic, and when is it justified?

Key findings

• The Bussgang decomposition provides an exact representation of non-linear system outputs as a scaled input plus uncorrelated distortion, even when the distortion is non-Gaussian and dependent on the input.
• For complex-valued Gaussian signals, the cross-correlation between the non-linear output and a second signal is proportional to the input cross-correlation, with the proportionality factor being the Bussgang gain $B = \mathbb{E}\{U(x)x^*\}/\mathbb{E}\{|x|^2\}$.
• In MIMO systems, the distortion vector $\bm{\eta}$ is generally correlated across entries, with the correlation matrix $\mathbf{C}_\eta = \mathbf{C}_z - \mathbf{B}\mathbf{C}_x\mathbf{B}^\mathrm{H}$, and this correlation is non-zero when the input signal covariance $\mathbf{C}_x$ is non-diagonal.
• Simulations show that at low ADC resolution (e.g., 4 bits), the off-diagonal elements of $\mathbf{C}_\eta$ are large and non-zero, indicating significant distortion correlation that cannot be ignored.
• At high ADC resolution (e.g., 8+ bits), the off-diagonal correlation coefficients in $\mathbf{C}_\eta$ are small and can be approximated as zero, justifying common simplifying assumptions in high-resolution systems.
• The generalized Bussgang decomposition for non-Gaussian inputs remains valid, with $\bm{\eta}$ uncorrelated with $\mathbf{x}$, but the distortion is not independent of $\mathbf{x}$, and its correlation structure must be carefully modeled for accurate performance analysis.