The Non-Linear Subcircuit

The noise in the non-linear part of the circuit is calculated by using the quasi-static approach, i.e. for every moment in time the voltages and currents are regarded as a time-dependend bias point. The noise properties of these bias points are used for the noise calculation.

Remark: It is not clear whether this approach creates a valid result for noise with long-time correlation (e.g. 1/f noise), too. But up to now, no other methods were proposed and some publications reported to have achieved reasonable results with this approach and 1/f noise.

Calculating the noise-current correlation matrix $(\underline{C}_{Y,nl})_{N\times N}$ needs several steps. The DC bias point taken from the result of the HB simulation is the beginning. Its values are the bias used to build the correlation matrix $(\underline{C}_{Y,DC})$. Each part is a $K \times K$ diagonal submatrix. The values are the power-spectral densities (PSD) for each harmonic frequency:

$$C_{Y,DC}(\omega_R) \qquad C_{Y,DC}(\omega_0 + \omega_R) \qquad C_{Y,DC}(2\cdot\omega_0 + \omega_R) \qquad \dots$$ (8.2)

where $\omega_R$ is the desired noise frequency.

The second step creates the cyclostationary modulation that is applied to the DC correlation matrix. The modulation factor $M(t)$ originates from the current power spectral density $S_i$ of each time step normalized to its DC bias value:

$$M(t) = \dfrac{S_i\left( u(t) \right)}{S_i(u_{DC})} = \dfrac{S_i\left( u(t) \right)}{C_{Y,DC}}$$ (8.3)

Note that this equation only holds if the frequency dependency of $S_i$ is the same for every bias, so that $M(t)$ is frequency independent. This demand is fullfilled for all practical models. So the above-mentioned equation can be derived for an arbitrary noise frequency $\omega_R$.

The third step transforms $M(t)$ into frequency domain. This is done by the procedure described in equation 7.13, resulting in a Toeplitz matrix.
The fourth and final step calculates the desired correlation matrix:

$$(\underline{C}_{Y,nl}) = M \cdot (\underline{C}_{Y,DC}) \cdot M^+$$ (8.4)