# Measurement: Practice Make Perfect – Part 2

by Pat Brown

This is a continuation of Measurement: Practice Makes Perfect – Part 1. I suggest viewing Part 1 prior to continuing with Part 2. As you may remember. Here’s a quick recap of what I did in Part 1:

1. Created a pulse in Wavosaur

2. Observed its “perfect” transfer function (flat magnitude and phase) in Arta

3. Delayed it with respect to time

4. Used group delay to locate it in time

This allowed the observation of the transfer function of a system with signal delay as though there were no delay. Of course delay is a fact of life with every physical system that you measure, so removing the latency or time-of-flight (TOF) is fundamental to audio and acoustic measurements. Some measurement systems do this for you automatically. Even so, it’s good to understand why it must be done.

I created the Dirac pulse in a wave editor. A “loopback” test performed on your sound card by your measurement system should yield essentially the same ideal response. The difference is that your time domain measurement system would likely use pink noise, sweeps, MLS or other stimuli. Regardless of how the impulse response is measured, we can consider that a signal with a much wider bandwidth than the device-under-test has been fed into it and collected at the other end. We can now see the carnage imparted by the system.

Band Limited Systems

In addition to adding delay, real-world systems are band limited. If you put an infinite range of frequencies in, you only get a finite range of frequencies out. This is not necessarily a bad thing. The human auditory system only needs about 3 decades of bandwidth, starting with 20 Hz, to fully assess the sonic character of an event. Anything more is wasted and can actually be detrimental. Do you really want AM radio signals to pass through your sound system?

The Dirac pulse of Part 1, our golden reference, will undergo some modification in Part 2. In a real-world system, the response would be high passed at the low frequency end since the response of a loudspeaker cannot extend to zero Hz (or DC). It this chapter I will assess the effect of a high pass filter used to band limit the system. I’ll keep the 1 ms TOF from Part 1. So, the signal is now delayed and band limited, which is a step closer to reality than a perfect pulse.

Let’s use Wavosaur to add a high pass filter. There are infinite possibilities in filter world, but I’ll select a Chebyshev 2nd-order and place it at 100 Hz. A second order filter rolls off at 12 dB/octave. The principles are the same with Butterworth, Bessel, etc. with some subtle differences in the resultant transfer function.

Note that the immediate effect in Wavosaur is that the impulse has changed shape slightly. Due to the presence of the filter, all frequencies are no longer the same level nor are they in phase. Let’s take it over to Arta to see the effect of the high pass filter in greater detail.
The time domain view is identical to the wave editor, but the power of a measurement platform vs. a wave editor is in the frequency domain. The frequency response magnitude clearly shows the high pass response. This is as expected. The group delay is much more interesting. The effect of the filter is to add delay, and this delay is mostly in the stop band of the filter, but extends well into the pass band. All loudspeakers are inherently high passed, so you will observe this behavior in any loudspeaker that you test. I’ve just simulated it with an electronic filter.

Group delay has given a quite nice picture of the effect of the high pass filter in a way that is easy to understand. The phase response of the transfer function reveals even more detail. Remember that to view the phase response, we must compensate for TOF or the phase will be wrapped due to the delay. From the group delay response, it is apparent that there is no longer a single compensation time. Stated another way, the amount of group delay (and phase shift) is now frequency-dependent. Filters do indeed complicate things.

Since the group delay appears to level out at high frequencies at 1 ms, let’s use that as the compensation. The movie shows how.
While the time-compensated group delay appears to be zero for most of the pass band of the system, the phase response reveals otherwise. Group delay provides a rather coarse resolution with regard to time. I’ll now use the phase response to “tweak” the delay compensation to a more precise value. The phase plot is the electron microscope of the audio practitioner.

Minimum Phase

But how much compensation? When is it “right?” The answer is right at our finger tips, and only a mouse-click away. Arta calculates and displays the “minimum phase” response of the system, based on the magnitude response of the transfer function. There exists an infinite number of possible phase responses for a given magnitude response. The minimum phase response reveals the least amount of phase shift that can be present for the given magnitude response. It is in fact calculated from the magnitude response. There is a great deal of information on minimum phase in the SAC Members Library. I’ll summarize it by saying that a minimum phase system has potentially higher fidelity than a system that exhibits a non-minimum phase response. Or, stated another way, it is a good thing to avoid introducing extra frequency-dependent delay into a system beyond what must exist based on the magnitude response.

I’ll use Arta to display the minimum phase response, and then compensate the arrival time until the measured phase response most closely matches it. Watch the movie.
So,  0.95 ms of delay compensation was required, a bit less than the 1 ms suggested by the group delay plot. This is the amount of time required to “zero” the phase response in the pass band. Note that the low frequency delay shows up as a slope in the phase response. Also note the sensitivity of the HF phase to the delay compensation adjustment. Even the smallest increment of 10 microseconds causes a large phase change. This is one of many reasons why we shouldn’t get too obsessed with “accurate” response beyond 10 kHz.
The minimum phase response of a system will
1. have the least slope (wrapping) of all possible phase responses, and
2. will pass through zero degrees at its center frequency (if properly polarized).
Burn those into your psyche because in the real world the phase response can get pretty crazy and you will need some truths to help interpret what you are seeing on the analyzer.

Conclusion

There are many lessons to learn from this simple example.

– All loudspeakers are inherently high passed. They are in effect filters and are measured as such.

– There will always be a frequency-dependent delay in the loudspeakers transfer function. A loudspeaker has no single “arrival time” and as such cannot be pinpointed to a single point in space. The ramifications for “time aligning” (or more correctly, “signal aligning”) are significant.

– Steeper filters produce more delay. This helps explain why various subwoofer designs can have similar magnitude responses, yet sound much different.

– A steep high pass filter added by a DSP to “protect” the subwoofer and shape the magnitude also smears its time response well above the frequency setting of the filter, potentially affecting the transient response of the system.

– If the “best” low frequency fidelity results from the least amount of group delay, then simpler is better.

An audio practitioner armed with a high resolution measurement system can fully assess the pros and cons of any loudspeaker design. Without measured data as a reference, you may never realize that the preferred “thonk” sound of woofer A as compared to woofer B is actually the result of the time smear caused by an 8th order high pass filter used to increase the power handling of the device. Much can be learned from investigating “why” you prefer one response over another.

In Part 3 I’ll go to the high frequency end of the spectrum and add a low pass filter, as the plot continues to thicken.   pb