Introduction

Figure 1: Linear system to be measured

Figure 1 depicts a linear system characterized by an impulse response h(n)h(n), driven by an input signal s(n)s(n), and producing the output signal r(n)r(n). The system identification problem is to estimate h(n)h(n) given known input/output signals s(n)s(n) and r(n)r(n). A practical method for identifying finite impulse responses is the swept-sine measurement technique, described below.

Sine Sweep Measurement Theory

In some cases, it is desirable to relax the power-maximizing constraint |s(n)|=1∀n|s(n)|=1∀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)s(n) to contain more energy at lower frequencies Entry 1. We might also be measuring a mechanical or acoustical system in which the motor controlled by s(n)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 2. The goal is to measure h(n)h(n), independently of the motor nonlinearity f(s)f(s). Performing the measurement is complicated by the fact that superposition no longer holds.

Figure 2: Hammerstein Model

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 s(n)s(n). This signal is a sine wave with a frequency that is exponentially increased from ω1ω1 to ω2ω2 over TT seconds Entry 2:

where K=ω1Tlnω2ω1K=ω1Tlnω2ω1 and L=Tlnω2ω1L=Tlnω2ω1. The MATLAB/Octave code generate_sinesweeps.m generates the appropriate sine sweep.

Now we consider how to extract the linearized impulse response from a measurement. In essence, we need to inverse filter the measurement by the excitation signal.

To this end, we realize that a useful property of s(n)s(n) is that the time delay ΔtNΔtN between any sample n0n0 and a later point with instantaneous frequency NN times larger than the instantaneous frequency at s(n0)s(n0) is constant:

This characteristic implies that after inverse-filtering the measured response, the signals due to the nonlinear terms in f(s)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)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)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.

Sine Sweep Measurement Procedure