Sine Sweep Measurement Theory

In some cases, it is desirable to relax the power-maximizing constraint $\vert s(n)\vert = 1 \forall n$ in favor of obtaining some other desirable measurement system properties. For example, we may care more about the accuracy of the measurement at lower frequencies compared to higher frequencies, so we would like the excitation signal $s(n)$ to contain more energy at lower frequencies [1]. We might also be measuring a mechanical or acoustical system in which the motor controlled by $s(n)$ behaves weakly nonlinearly. If the nonlinearity is memoryless and is NOT preceded by any filtering, then the system to be measured matches the Hammerstein model shown in Figure 11. The goal is to measure $h(n)$, independently of the motor nonlinearity $f(s)$. Performing the measurement is complicated by the fact that superposition no longer holds.

Figure 11: Hammerstein Model

Mathematically, the Hammerstein system behaves as follows:

$$r(n) = (f(s) \ast h)(n)$$ (6)

It turns out that we can obtain both of these desirable measurement system properties by using a new excitation signal $s(n)$. This signal is a sine wave, whose frequency is exponentially increased from $\omega_1$ to $\omega_2$ over $T$ seconds [2].

$$s(n) = \sin [ K(e^{-n/Lf_s} - 1) ]$$ (7)

where $K = \frac{\omega_1T}{\ln \frac{\omega_2}{\omega_1} }$ and $L = \frac{T}{\ln \frac{\omega_2}{\omega_1} }$. The MATLAB/Octave code generate_sinesweeps.m generates the appropriate sine sweep.

The important property of $s(n)$ is that the time delay $\Delta t_N$ between any sample $n_0$ and a later point with instantaneous frequency $N$ times larger that the instantaneous frequency at $s(n_0)$ is constant:

$$\Delta t_N = T\frac{\ln (N)}{\ln \frac{\omega_2}{\omega_1}}$$ (8)

This characteristic implies that after inverse filtering the measured response, the signals due to the nonlinear terms in $f(s)$ are located at specific places in the final response signal. Consequently, the linear contribution to the response, which is proportional to $h(n)$ 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 $s(n)$ 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.