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.
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.
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.
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.
– 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.
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