Example 2: DCA of a non-linearity followed by its inverse

To clarify the interpretation of the different distortion contributions obtained with the BLA-based DCA, we consider the trivial example of a static non-linearity followed by its inverse (Figure E2.1). The first non-linear block is an exponential function and the second its inverse: a logarithm. The cascade of both blocks results in a perfectly linear system.

Figure E2.1 Cascade of two non-linear systems studied in this example.

The system is excited by random-phase multisines with an $f_{0}\!=\!1\mathrm{Hz}$ which excite all frequencies up to $100\mathrm{Hz}$. The RMS of the multisine was set to $0.5\mathrm{V}$. The steady-state spectrum of the signal $I$, measured between the two non-linear blocks, is shown in Figure E2.2.

Figure E2.2 Spectrum of the signal $I$ between the non-linear blocks.

The frequency bins excited by the multisine are shown in black, the remaining frequency lines in magenta. The measured distortion power at the internal signal is shown with the magenta line in the plot. The amount of non-linear distortion at the intermediate signal in this cascade is very high (signal to distortion ratio of $10\mathrm{dB}$), but all the distortion is completely cancelled out by the second block, so that the input and output signals are exactly the same. We calculate the BLAs of the two non-linear blocks using (2.4)-(2.6). $10^{4}$ different-phase multisines were simulated to obtain an adequately low uncertainty on the BLA-estimates in this strongly non-linear circuit. The obtained BLAs and their $3\sigma$ uncertainty bound are shown in Figure E2.3.

Figure E2.3 BLAs of the two non-linear blocks in the cascade.

With these BLAs, the covariance matrix of the distortion sources can be calculated using (3.7). In this simple example, there are two distortion sources, one for each non-linearity. This results in a $2\times2$ covariance matrix $\mathbf{C}_{\mathbf{D}}$. The elements on the diagonal of $\mathbf{C}_{\mathbf{D}}$ describe the power of each of the distortion sources. The off-diagonal elements indicate the correlation between both sources. The values of the distortion covariance matrix are shown in Figure E2.4.

Figure E2.4 Elements of $\mathbf{C}_{\mathbf{D}}$ as a function of frequency.

The three distortion contributions to the output can now be calculated. The FRF from each distortion source to the output can be obtained using (3.5). Here, we have:

With $\mathbf{C}_{\mathbf{D}}$, $T_{_{\left[1\right]}}$ and $T_{_{\left[2\right]}}$, we can calculate the distortion contributions to the output of the circuit. The direct distortion contributions due to the first stage is:

The direct distortion contributions due to the second stage is:

The correlation distortion contribution is given by:

The obtained distortion contributions are shown in Figure E2.5. The two direct contributions are equal in amplitude and both positive. The correlation contribution is equal to the sum of the two direct contributions, but opposite in sign. The sum of all contributions (shown in gray on the plot above) therefore lies very close to zero.

Figure E2.5 Distortion contributions obtained for this example.

With this very simple example we have shown the effectiveness of the BLA-based DCA to predict the distortion contributions of a strongly non-linear circuit under a modulated excitation signal. Additionally, we have shown that it is important to keep the correlation distortion contributions into account to obtain a correct result.