0.2 Measuring impulse responses using golay complementary sequences  (Page 2/2)

1. Sound interfaces attenuate frequencies near DC and near half of the sampling rate $f_S/2$ . [link] shows the measured magnitude response of the sound interface. Ideally it would beconstant for all frequencies. However, [link] shows that the -3dB point for the DC blocker lies near 5Hz. The anti-aliasingfilters cause the magnitude to roll off sharply such that the high frequency -3dB roll-off point is about 20.9kHz.

   

2. The same sound interface filters cause the phase response measurement to be incorrect as well. Although ideally the phaseresponse would be 0 radians for all frequencies, [link] reveals that the phase measurement is distorted in roughly the same regions where the magnitude measurement is distorted.

   

3. Even when a computer is programmed to pass an input signal flowing into the sound interface input out of the sound interfaceoutput as fast as possible, there is a delay or latency . This delay is typically on the order of tens of milliseconds. [link] shows the impulse response measured on the PreSonus card. The latency is approximately 62ms.The latency could have been decreased some by adjusting software settings in pd. The impulse response also rings noticeably due to the anti-aliasing filters.

   

   For reference, we provide the measured impulse response shown in [link] .

Minimum phase systems

One further consequence of the delay is that determining the phase response of the measured system is more complicated. The delay isresponsible for a linear phase term since $\delta (n-k)\leftrightarrow e^{-j2\pi fk/f_S}$ . If the delay is known (or measured), then it may be removed by multiplying the measured spectrumby $e^{j2\pi fk/f_S}$ . However, if the system being measured is known to be minimum phase, then the folded-cepstrum method may be applied to find the minimum phase frequency response corresponding to the measured frequency response.

The transfer function measurement toolbox assumes that the system being measured is minimum phase. This is a valid assumption in manycases. For instance, all strictly positive real transfer functions are minimum phase. Dissipative systems are strictly positive real(and therefore minimum phase) if the appropriate quantity is measured and the sensor and motor are collocated. For example, if $s\left(n\right)$ controls a motor exerting a force on a dissipative system, and $r\left(n\right)$ is the velocity at that same point, the corresponding transferfunction will be minimum phase. This holds for other dual variable pairs such as torque and angular velocity, voltage and current, andpressure and fluid flow.

For systems that are not minimum phase, such as systems involving a transmission delay between the input and output quantities, the phaseplotted by the transfer function measurement toolbox is not the system phase response, but rather the minimum phase response corresponding tothe measured system phase response.

High pass filter measurement

The circuit shown in [link] was measured using the Golay complementary sequence method to show how the sound interface non-idealities affect a measurement. $V_{IN}$ was connected to the output of channel 1 of the PreSonus sound interface,and $V_{OUT}$ was connected to the line input of channel 1 of the interface.

The analog transfer function $H\left(f\right)$ can be determined analytically using the voltage divider rule:

In this case, $R=1$ k $\Omega$ and $C=0.47\mu$ F, so the -3dB point is about $f_{3dB}=\frac{1}{2\pi RC}\approx 340$ Hz. [link] and [link] show that the frequency response is accurately measured in the range of about 10Hz to about $9f_S/20$ .

The ringing in the measured impulse response distracts from the more subtle characteristics of the ideal high pass filter impulse response. For transferfunctions that pass large amounts of energy at high frequencies, it may be more instructive to inspect the frequency domain measurement results.

Sine sweep measurement of a weakly nonlinear loudspeaker driver

To exeraggerate the nonlinearity of a loudspeaker, we cut the cone of a mishandled driver as shown in [link] . We monitored the sound pressure several centimeters in front of the dustcap using an AudioTechnica AT4049a microphone, which has a flat magnitude response to within 3dB from 100Hz to 5kHz. The output from channel 1 of thePreSonus sound interface was connected to the speaker via a power amplifier, and the microphone was connected to the microphone input ofchannel 1 on the sound interface. The following results are typical of sine sweep measurements to show how with a weakly nonlinear motor.

Inverse filtering the measured response results in [link] , which is a plot of nonlinear2ImpResp.wav . The linear contribution corresponds to the spike at the beginning,while the weakly nonlinear terms are clustered closer to the end of the response.

The main linear contribution is cut out and plotted in [link] . The measurement was not made in an anechoic chamber, so there is a reflection about 15ms after the mainimpact.

The nonlinear terms are shown magnified in [link] . The lower order nonlinear terms toward the right have larger magnitude but overlap less in time (see [link] ). Note that [link] implies that the overlapping could be reduced by increasing the total length $T$ of the sweep excitation signal.

The magnitude and phase responses corresponding to the linear impulse response term from [link] are shown in [link] and [link] in blue . For comparison, another sine sweep measurement was made at a lower levelso that the speaker behaved approximately linearly. Decreasing the level also resulted in more noise and even some systematic error, asis evidenced by the red curves in [link] and [link] . This comparison demonstrates that making measurements at larger levels can reduce the effects of noise, whilenonlinear motor effects can be overcome with the sine sweep measurement technique.