Appendices

In this appendix, we demonstrate how the BLA and non-linear distortion change in function of the power of the input multisines and how both can be different from the results obtained with a single-tone or two-tone excitation signal. Consider the following static non-linearity $y = \text{tanh}(u)$ which is a saturation function that closely resembles the voltage-to-current transfer function of a bipolar transistor [35] or a MOST differential pair biased in weak inversion [36]. The validity of the harmonic distortion scaling rules in function of the input power is verified by applying a single-tone, two-tone and random-odd RPM (Section 2.1) to this weakly nonlinear function (Figure A.1). Even though the different excitations result in a different absolute value for the distortion component, this specific example shows that for static weakly non-linear systems the same scaling rules for $\mathrm{2^{nd}}$-order distortion (1 dBW/dBW) and $\mathrm{3^{th}}$-order distortion (2 dBW/dBW) apply until compression occurs.

In a more realistic setting (low-noise amplifier in 90 nm CMOS), large discrepancies in the inter-modulation distortion between two-tones and multisines were documented (Figure A.2) [16]. In the case of the two-tone, a local minimum (the so-called sweet-spot) appears which gives a wrong impression about the actual inter-modulation distortion present in the circuit when complex modulated signals are used.

Figure A.1 Applying different excitation signals to a weakly nonlinear system $y = \text{tanh}(u)$ reveals that different harmonic distortion levels are obtained.

Figure A.2 The inter-modulation distortion reveals a clear local minimum with a two-tone excitation. This minimum dissapears completely when a multisine excitation is used. The figure is adapted from [16].

In the circuit shown in Figure 3.1, the following equations describe the behaviour of the input, output and feedback dynamics:

The relation between $\mathbf{U}$ and $\mathbf{Y}$ is given by the BLA:

Plugging equation (A.1) into (A.3) and solving for $\mathbf{Y}$, we obtain:

Using this expression in (A.2) and grouping the terms in $R$ and $\mathbf{D}$ yields equation (3.5).

To calculate the contributions of the distortion sources to the output of the circuit, an algorithm similar to the one described in [25] is used. The different S-matrices of the components in the circuit shown in Figure 4.1 are gathered in a matrix $\mathbf{T}$, while the distortion sources are gathered in a vector $\mathbf{N}$:

where $\Gamma_{\mathrm{in}}$ and $\Gamma_{\mathrm{out}}$ are the reflection factors presented to the circuit by the reference source and load respectively. $\mathbf{S}_{\mathbf{A}\rightarrow\mathbf{B}}^{\mathrm{BLA}}$ is the block diagonal matrix of size $P\!\times\!P$ which contains the BLAs of the circuits as defined in equation (4.1). $\mathbf{D}$ is the vector of distortion sources of length $P$ defined in the same expression. $\mathbf{P}$ is the S-matrix of the package defined in Figure 4.1 of size $\left(P+2\right)\!\times\!\left(P+2\right)$ for a circuit with $2$ external ports.

The interconnection between the different parts of the circuit is represented by the following matrix:

The incident-waves at all ports generated by the sources in $\mathbf{N}$ is given by the following expression:

Since we are only interested in the wave incident to the load, just the second row of $\mathbf{W}^{-1}$ is used. Also, the first $P+4$ elements of $\mathbf{W}^{-1}$ can be ignored, because the first $P+4$ elements of $\mathbf{N}$ are zero. This finally leads to the expression for $\mathbf{T}_{\mathrm{out}}$ used in equation (4.3):

Predicting the frequency response from the main reference signal to the input and output waves at the ports of the sub-circuits is done using the same matrix $\mathbf{W}^{-1}$ as was used in Appendix 3. but now, the $\mathbf{N}$-vector is set to the following:

where $Z_{0}$ is the chosen reference impedance and $\Gamma_{S}$ is the reflection factor presented by the reference source. All A-waves in the circuit are now predicted by (A.4). The frequency response from the reference voltage source to the A-waves radiating into the sub-circuits are found at $\left[\mathbf{A}_{\mathrm{all}}\right]_{4+P+1..4+2P,1}$. The frequency response from the reference voltage source to the B-waves at the ports of the sub-circuits are found at $\left[\mathbf{A}_{\mathrm{all}}\right]_{5..5+P,1}$.