3. Distortion Contribution Analysis & BLA

By fixing the input signal class (fixed PSD and fixed PDF ) and working with the BLA framework, the non-linear distortion in a circuit can be treated as if it were noise. Combining the BLA with a noise analysis then allows one to determine the dominant source of non-linear distortion in a system. The basic idea is simple [12][22], but a rigorous treatment of the concept has not been detailed in literature. The main difference between the noise analysis in a BLA-based DCA and the classic noise analysis is that all distortion sources are correlated. Taking this correlation into account is very important to obtain the correct result for the DCA and is one of the main contributions of this paper.

Figure 3.1 The general system under consideration consists of multiple non-linear systems in a feedback configuration.

Consider $N$ SISO non-linear systems embedded in a linear feedback structure as is shown in Figure 3.1. The whole system is excited by random-phase multisines $R$ with a specified PSD and PDF. Using (2.2), the output of the system at the $k^{\mathrm{th}}$ bin of the DFT can be written as:

This expression indicates that the output contains a best linear contribution to the input ($G_{R\rightarrow Y_{t}}^{\mathrm{BLA}} R$) and a distortion term $D_{t}$. The goal of the DCA is to write $D_{t}$ as the sum of contributions stemming from the $N$ non-linear blocks in the circuit. As explained in the previous section, $D_{t}\left(k\right)$ has noise-like properties which means that only the power of the output distortion, or $\mathbb{E}\left\{ D_{t}\left(k\right) D_{t}^{\mathsf{H}}\left(k\right)\right\}$, can be considered. Although (3.1) also holds for systems excited by single-tone and two-tone signals, the distortion term $D_t(k)$ for these deterministic signals exhibits different statistical properties compared to noise excitations (Appendix 1). At present, there does not exist a method which relates the distortion term obtained with noise excitations to the one obtained with deterministic signals. As a result, the proposed BLA-based DCA is currently only applicable to noise excitation signals.

To determine the distortion contributions separately, first consider the BLAs of the different non-linear systems. All inputs and outputs of the non-linear sub-circuits are gathered frequency by frequency in column vectors $\mathbf{U}\left(k\right)$ and $\mathbf{Y}\left(k\right)$:

where $Y_{\left[n\right]}\left(k\right)$ and $U_{\left[n\right]}\left(k\right)$ indicate the output and input DFT spectra of the $n^{\mathrm{th}}$ sub-circuit respectively. The different SISO BLA, as defined in (2.1), are grouped in a diagonal matrix:

The input-output relation of all non-linear systems can now be written simultaneously as follows:

where $\mathbf{D}\in\mathbb{C}^{N\times1}$ contains the non-linear distortion introduced by the $N$ sub-systems. From here, the frequency indices $\left(k\right)$ and $\left(j\omega_{k}\right)$ will be omitted for notational simplicity. It is shown in Appendix Appendix 2 that the output signal $Y_{t}$ of the total system can be written as:

where $\mathbf{B}$, $\mathbf{A}$ and $\mathbf{M}$ are the linear blocks connected to the non-linear sub-circuits as shown in Figure 3.1. $\mathbf{I}_{N}$ is the identity matrix of size $N$. The second part of (3.5) yields the expression for the output distortion as a function of the distortion in the sub-systems. Considering the power of the distortion at $\left(k\right)$:

with $\mathbf{T}_{\mathrm{out}}=\mathbf{B}\left(\mathbf{I}_{N}+\mathbf{G}_{\mathbf{U}\rightarrow\mathbf{Y}}^{\mathrm{BLA}}\mathbf{M}\right)^{-1}$ a row vector of length $N$ that contains the Frequency Response Function (FRF) from each distortion source to the output. The $n^{\mathrm{th}}$ element of $\mathbf{T}_{\mathrm{out}}$ will be called $T_{\left[n\right]}$ from now on. $\mathbb{E}\left\{ \mathbf{D}\mathbf{D}^{\mathsf{H}}\right\} =\mathbf{C}_{\mathbf{D}}$ is the covariance matrix of the distortion introduced by the non-linear sub-systems.

A two-step procedure is used to obtain an estimate of $\mathbf{C}_{\mathbf{D}}$, which is similar to the way we determined the BLA itself in section Section 2 [19]. First, the covariance matrix of the stacked input-output vectors $\mathbf{C}_{\mathbf{Z}}$ is determined as in equation (2.5), but now $Y^{\left(m\right)}$ and $U^{\left(m\right)}$ are replaced by the stacked input and output signals defined in (3.2). $\mathbf{C_{\mathbf{Z}}}$ is multiplied by the number of different-phase multisines $M$, as we are interested in the power of the distortion, rather than in the uncertainty on the BLA-estimate. This $\mathbf{C}_{\mathbf{Z}}$ is now a $2N \times 2N$ matrix. To obtain a full-rank estimate of $\mathbf{C}_{\mathbf{Z}}$, the response to at least $2N$ different-phase multisines must be simulated. $\mathbf{C}_{\mathbf{D}}$ is then calculated starting from $\mathbf{C}_{\mathbf{Z}}$ in the following way:

The matrix product in (3.6) can be re-written as:

herein $\mathbf{C}_{\mathbf{D}}$ is an Hermitian matrix. The complex conjugate contributions of $\left[\mathbf{C}_{\mathbf{D}}\right]_{i,j}$ and $\left[\mathbf{C}_{\mathbf{D}}\right]_{j,i}$ will therefore combine to form a single, real-valued distortion power contribution. The expression for the distortion at the output can now be simplified as follows:

Equation (3.8) contains all the different distortion contributions: each element of the covariance matrix of the distortion sources is transferred to the output. The total distortion at the output is then the sum of all these contributions. The contributions can be sorted according to their magnitude to determine the dominant distortion contribution.

From now on, we will refer to the distortion contributions due to the diagonal elements of the distortion covariance matrix as direct distortion contributions ( $C_{_{\left[i\right]}}$ ). The contributions due to the off-diagonal elements will be called correlation distortion contributions ( $C_{_{\left[i,j\right]}}$ ).