# BLA-based Design and Analysis of VCO-based Sigma-Delta Modulator

Nonlinear system theory is involved. The available theoretical frameworks depend on the assumed system properties and the applied signals. An example is the Volterra theory framework which uses a series expansion of the systemâ€™s behaviour around an operating point. This operating point can be fixed (classical Volterra theory), periodically varying, or can consist of an expansion around a reference signal. Volterra series expansions have the advantage to be a direct extension of the linear system theory, but have the disadvantage that they often lead to complex expressions and that they can only describe weakly nonlinear systems.

Various techniques exist to reuse the linear frameworks in system theory, system design, identification, and control using approximative models of the nonlinear system. Examples of such techniques include (but are not limited to) the Describing Functions (DF) and the Best-Linear Approximations (BLA) framework.

The DF framework allows to approximate the input-output relationship of static nonlinear blocks (including saturation and/or hysteresis phenomena) by a linear gain which is function of the excitation signalâ€™s characteristics. The DF considers a class of excitations which are combinations of input signals comprising a constant operating point, a single-sinusoid input (SIDF), a two-sinusoid input (TIDF) or a random input (RIDF). Furthermore, the DF assumes that the filtering characteristic of the overall system filters signals such that the output signal assumptions are a good approximation of the reality.

The theory around the DF allows the analysis/design of not only (nonlinear) control loops, but also the study of limit cycles within autonomous nonlinear systems. Throughout the years, the DF has been proven beneficial for the nonlinear analysis of oscillators [11, 12, 13] and sigma-delta modulators. However, the DF has two main drawbacks: its potential to measure frequency dynamics is limited and it does not give insight in the (unmodelled) residuals/nonlinear distortion.

The Best-Linear Approximation framework almost starts from the dual viewpoint: it approximates the linearised dynamic behaviour (in least-squares sense) from a (strongly) nonlinear system assuming a Gaussian-like input signal. Different BLA extraction techniques exist (robust method, fast method and local polynomial method) using multisine excitations (a subset of Gaussian-like inputs) which enable the measurement of the linear dynamic behaviour under a wideband excitation, and allows the separation of the steady state and transient response, additive measurement noise and even/odd nonlinear distortions. Lately, the BLA has been succesfully extended towards non-autonomous (periodically) time-varying systems. Using the BLA and the characterized distortion levels, a Distortion Contribution Analysis (DCA) can be performed to pinpoint the dominant sources of nonlinear behaviour. The disadvantage of exclusively working with Gaussian-like signals is the impossibility to analyse (periodically) autonomous systems.

It can be concluded that the BLA and the DF start from a different point of view and result into complementary application domains. In particular cases, the BLA and DF are related to each other. However, no general theory exists for the linear approximation for both autonomous and non-autonomous nonlinear dynamical systems, which combines concepts from both the BLA and DF. The aim of this PhD is to develop and illustrate such a theory to more effectively analyse and design complex nonlinear systems.