Mathematically, the Hammerstein system behaves as follows:
It turns out that we can obtain both of these desirable measurement system properties by using a new excitation signal . This signal is a sine wave, whose frequency is exponentially increased from to over seconds .
where and . The MATLAB/Octave code generate_sinesweeps.m generates the appropriate sine sweep.
The important property of is that the time delay
between any sample and a later point with instantaneous frequency
times larger that the instantaneous frequency at is constant:
This characteristic implies that after inverse filtering the measured response, the signals due to the nonlinear terms in are located at specific places in the final response signal. Consequently, the linear contribution to the response, which is proportional to can be separated from the other nonlinear terms. We can thus measure a linear system even if it is being driven by a weakly nonlinear motor.
Because the frequency of increases exponentially, the system is excited for longer periods of time at lower frequencies. This means that the inverse filter averages measurements at lower frequencies longer, so this measurement technique is better suited to especially low-pass noise sources.