1. Introduction

The decrease of supply voltages in aggressively scaled technologies results in non-linear distortion to become one of the main limiting factors for the dynamic range of analog electronic circuits. Still, the distortion is often taken into account at later stages of the design only by using a single- or two-tone test. The total harmonic distortion or intermodulation distortion is then intended to describe the non-linearity of the circuitry. These numbers give an indication of the total distortion without providing in-depth insight into its origin.

The aim of the DCA is to split the total distortion into contributions of each sub-circuit [1]. Comparing the different contributions allows the designer to pinpoint the dominant sources of non-linear distortion and hereby effectively reduce the total distortion [2]. Note that, taken from a bird’s eye view, the DCA closely resembles a noise analysis. The difference is that it is now applied to non-linear distortion sources.

The first DCA methods were based on Volterra theory [3] [1] [4]. The method has been illustrated on small circuits and has been extensively used in both the analysis and the design of electronic circuits [1]. However, the original Volterra-based DCA had some severe limitations:

• Only weakly non-linear circuits could be analysed. In strongly non-linear circuits, the Volterra series obtained around the DC operating point fail to converge, which limits the use of the DCA.
• Only smaller circuits could be analysed. The number of Volterra distortion contributions rises quickly for larger circuits. A simple Miller op-amp, for example, yields over 700 contributions [1], making interpretation of these results more difficult, if not impossible.
• Only the distortion under single-tone or two-tone excitation signals had been considered. Exciting a circuit with practical complex modulated excitation signals, however, has a big influence on the non-linear behaviour of the circuit and hence on the distortion (in Section 3, we demonstrate this difference on an example)
  [5].

Many of the limitations of the original Volterra-based DCA have been overcome in more recent years. The phasor method and its variations simplify the obtained distortion expressions such that more complex circuits can be intuitively analysed [4] [6] [7] [8].
Alternatively, state-space approaches were introduced to circumvent the complexity explosion of the Volterra distortion contributions for larger circuits [9] [10]. Furthermore, an extension to strongly non-linear circuits has been proposed in [11]. Unfortunately, all these techniques mainly capture the distortion generated by single-tone or two-tone excitations, and often require circuit-specific assumptions which prevent the general applicability of these techniques.

More recently, the BLA has been used to perform a DCA on analog electronic circuits. The idea was originally proposed in [12] and has been applied to several examples in the past [13] [14] [15]. In the BLA framework, the behaviour of a non-linear system is approximated in least-squares sense by a linear system. As a consequence, the distortion introduced by the system can be represented by an additive noise source. Combining the BLA analysis with a classic noise analysis yields a DCA which solves some of the drawbacks of the classic Volterra-based implementations, at the cost of an increased simulation time. The main benefits of the method are:

• Linear models are used to describe the dynamic behaviour of the sub-circuits, while the distortion in the circuit is represented by noise-like sources. The concept of linear dynamic systems and noise are familiar to all designers.
• The analysis also applies to modulated signals, which leads to an accurate and realistic representation of the non-linear distortion generated by the circuit in real operation.
• The BLA method does not require simplified device models or accessibility to internal nodes of the device models.
• The validity of the BLA is not restricted to weakly non-linear circuits. Strongly non-linear power amplifiers and hard saturation can still be modelled with the BLA. However, in this paper, we rule out strongly non-linear circuits designed for frequency translation, like mixers, phase-locked loops …

All previous implementations of the BLA-based DCA use a simplified representation of the circuit, ignoring possible correlation of the distortion introduced by different stages on one hand and input-output impedances of the circuit on the other hand[12] [16] [13] [14]. In this paper, we link the BLA-based DCA to the theoretical framework of the BLA [17] [18] [19] (Section 2 and Section 3). A first contribution is to correctly take the correlation between different distortion sources present in the circuit into account. Secondly, the BLA-based DCA is extended to the use of S-parameters to represent the sub-circuits (Section 4). This extension takes reverse gain and terminal impedances of the sub-circuits into account, which enables a BLA-based DCA at the transistor level. The introduction of S-parameters moves the sub-circuit representation from Single-Input Single-Output (SISO) sub-blocks to Multiple-Input Multiple-Output (MIMO) sub-blocks, which complicates the identification of the BLA. Section 5 details the simulations required to estimate the BLA of the MIMO sub-circuits correctly. Finally, the BLA-based DCA is applied to a two-stage Miller op-amp, a Doherty power amplifier and a Gm-C biquad to show the benefits and general applicability of the method (Section 7).