Demodulation

Demodulation extracts a phasor from a steady-state time signal at selected multiples of the drive frequency. It is the response measure to use when you want quantities comparable to a lock-in amplifier or vector network analyzer: amplitude, phase, and response frequency.

The inputs are modal displacement time series \(x_i(t_n)\). Demodulation is applied directly to those modal coordinates and, optionally, to a weighted total response.

For each selected multiple \(\mu_m\), the demodulation frequency is

\[ \omega_m = \mu_m \omega_d, \]

with \(\omega_d\) the drive angular frequency.

For each mode \(i\), the complex coefficient is

\[ C_{m,i} = \sum_{n=1}^{N_t} w_n \, x_i(t_n) \, e^{-j \omega_m t_n}, \]

where \(w_n\) is the chosen window. The window coherent gain is

\[ c_g = \sum_{n=1}^{N_t} w_n. \]

Amplitude and phase then follow as

\[ A_{m,i} = \frac{2 |C_{m,i}|}{c_g}, \qquad \phi_{m,i} = \arg(C_{m,i}). \]

If you pass \(\text{modal_contributions} = (\phi_1, \dots, \phi_{n_{\mathrm{modes}}})\), the total response is formed from the complex coefficients,

\[ C_m^{\mathrm{total}} = \sum_{i=1}^{n_{\mathrm{modes}}} \phi_i \, C_{m,i}. \]

This is important: Poscidyn combines the modal coefficients before taking magnitude and phase, which preserves interference between modes.

If modal_contributions represents a mode-shape vector evaluated at a measurement point, the "total" block corresponds to the physical response at that point. If it is omitted, unit weights are used.

Parameters

multiples: non-empty sequence of frequency multiples. (1.0,) extracts the driven component, (1.0, 2.0, 3.0) adds superharmonics, and (1/3,) extracts a subharmonic.
window: optional analysis window. Supported values are None, "hann", and "hamming".
modal_contributions: optional 1D weight vector of length n_modes used to construct the total response.