2. The Single-Input Single-Output Best Linear Approximation

Instead of working with deterministic input signals, such as a sine wave or a two-tone, the BLA framework considers noise excitation signals with a fixed Power Spectral Density (PSD) and Probability Density Function (PDF). Examples are filtered white Gaussian noise and telecommunication signals with a specified bandwidth [20]. When the excitation signals from the specified class of signals are applied to a Period-In Same Period-Out (PISPO) non-linear system, the response of the system can be approximated in least-squares sense by the BLA [17]. Consider a SISO non-linear system with input $u\left(t\right)$ and output $y\left(t\right)$ placed in a feedback configuration (Figure 2.1).

Figure 2.1 The BLA framework allows one to model non-linear systems as the combination of a linear system $G_{U\rightarrow Y}^{\mathrm{BLA}}$ and a noise-like source $d(t)$ which represents the non-linear distortion.

The whole system is excited by a reference signal $r\left(t\right)$ with a fixed PSD and PDF, the corresponding BLA of the system is then defined as:

where $S_{yr}$ and $S_{ur}$ are the cross-power spectrum between the reference signal $r\left(t\right)$ and the output $y\left(t\right)$ and input $u\left(t\right)$ respectively [17] [18]. $\mathcal{F}\left\{ x\left(t\right)\right\}$ represents the Fourier transform of $x\left(t\right)$ and the expected value operator $\mathbb{E}\left\{ \bullet \right\}$ is taken with respect to the random reference signal $r\left(t\right)$.

The difference between the actual output $y\left(t\right)$ of the non-linear system and the output predicted by the BLA is denoted by $d\left(t\right)$. The distortion term $d\left(t\right)$ is zero mean, uncorrelated with the reference signal $r\left(t\right)$ and behaves like noise [18]. In the frequency domain, the input-output relation at each excited frequency bin $k$ is written as:

wherein $Y\left(k\right)$ and $U\left(k\right)$ are the DFT spectra of $y\left(t\right)$ and $u\left(t\right)$ respectively, evaluated at the $k^{\mathrm{th}}$ frequency bin. Equation (2.2) is the key expression that allows to use the BLA in a DCA. It tells that the output of each sub-circuit can be written as the sum of the output of a signal-dependent linear dynamic circuit and an additive noise source $D\left(k\right)$ representing the distortion. Calculating the frequency response function from each distortion source to the considered output of the total circuit allows one to compute the different distortion contributions.

Instead of working with noisy excitation signals directly, Random Phase Multisine (RPM) are commonly used to estimate the BLA for a Gaussian input signal. A RPM is a sum of harmonically related sine waves with a random phase:

where $f_{0}$ is the base frequency of the multisine. $A_{k}$ and $\phi_{k}$ are the amplitude and phase of the $k^{\mathrm{th}}$ tone in the multisine. If the phases are drawn randomly from a uniform distribution $\left[0,2\pi\right[$, the multisine PDF converges to a Gaussian PDF when a large number of frequencies $N$ is considered. The amplitude coefficients $A_{k}$ in the multisine can be chosen in a deterministic way to set the required PSD. In RF applications, the multisine only excites frequency bins around a centre frequency $f_{\mathrm{c}}$ between a minimum frequency $f_{\mathrm{min}}$ and a maximum frequency $f_{\mathrm{max}}$. $f_{\mathrm{min}}$, $f_{\mathrm{max}}$ and $f_{\mathrm{c}}$ are all set to integer multiples of the base frequency $f_{0}$. In baseband applications, lowpass multisines are used which excite frequencies starting from DC, so $f_{\mathrm{min}}$ is equal to $f_{0}$ in that case.

To separate even and odd non-linear distortion contributions in a baseband circuit, odd lowpass multisines are commonly used ($A_{k}\!\!=\!\!0$ for even $k$). An even non-linearity always combines an even number of frequencies in the multisine, so its distortion contributions will fall on the even frequency bins. An odd non-linearity will only return contributions on odd frequency bins so, just by design of the excitation signal, the even and odd non-linear contributions generated in the circuit are separated.

The odd non-linear distortion will end up on the excited frequency lines, which complicates estimating the amount of odd-order distortion between $f_{\mathrm{min}}$ and $f_{\mathrm{max}}$. To overcome this issue, random-odd RPM are used [21]. In a random-odd RPM, one odd excited line is left out randomly out of groups of three. On these detection lines, as these non-excited frequency bins are commonly called, an estimate of the odd non-linear distortion is easily obtained. The use of such a Random-odd RPM is illustrated in the first example.

We introduce the “robust method” to determine the BLA of a circuit as it allows to estimate the BLA and the distortion in the circuit with the highest accuracy [21]. $M$ different-phase multisines are applied to the system. In those different-phase multisines only the $\phi_{k}$ are changed in (2.3), the amount of tones ($N$) and the amplitude of the tones ($A_{k}$) is kept the same for each multisine. The steady-state response of the circuit to each of the different-phase multisines is then determined using a large-signal simulation. The steady-state spectrum of the reference signal, input signal and output signal of the circuit under excitation by the $m^{\mathrm{th}}$ different-phase multisine is labelled $R^{\left(m\right)}$, $U^{\left(m\right)}$ and $Y^{\left(m\right)}$ respectively.

The BLA of the system is now obtained in a two-step procedure. First, the Single-Input Multiple-Output (SIMO) BLA from the reference signal to the stacked output-input vector ($\mathbf{Z}$) is determined by averaging over the steady-state response to the different-phase multisines:

wherein $\mathbf{C}_{\mathbf{Z}}$ is the sample covariance matrix of $\mathbf{Z}$, expressing the uncertainty on the estimate. $\bullet^{\mathsf{H}}$ indicates the Hermitian transpose. Finally, the BLA of the system operating in feedback is determined as:

Furthermore, the uncertainty on the BLA-estimate can be calculated as:

More details about the experiments and algorithm needed to determine the BLA are given in Section Section 5 and in references [17] [19] [18].

The estimate of the uncertainty on the BLA is used to determine the number of different-phase multisines needed to obtain a sufficiently certain estimate of the BLA. When the uncertainty is too high for the specific application, more different-phase multisines are simulated and added to the set of signals until a sufficiently low uncertainty is obtained. In strongly non-linear circuits, it can take several hundreds of different-phase multisines to obtain a good estimate as the standard deviation only decreases with the square-root of this number.