by Pat Brown
One thing that I stress to new users of modern measurement systems is the importance of practice. You don’t want to hit the streets with one of these if you haven’t first nailed down how to use it. Unlike sound level meters and real-time spectrum analyzers, you don’t just turn it on and observe the response. Fortunately, you can learn a lot of what you need to know without spending a dime or even making a measurement. I’ll use a few downloadable applications to cover some basics.
Most acoustic tests are based on the impulse response. But, no balloon pops or hand claps here – we’ll create an impulse using a wave editor. Wavosaur is a freeware wave editor and one of my favorites. I’ll start by creating a blank WAV file and inserting a “Dirac pulse” at time zero. This movie shows the steps.
Now it’s time to fire up the measurement program. I’ll use Arta to open the WAV file just created. Once opened, it can be transformed to the frequency domain for spectral analysis. This movie shows the steps.
A perfect pulse in the time domain has a perfectly flat magnitude and phase in the frequency domain. Such a system, if it existed, would pass a square wave without distortion. Take a good look, because you will never see a real-world device that exhibits this perfect response. Let’s add a dose of reality.
It takes time for a signal to pass through a system. Our impulse is at time zero. I’ll use Wavosaur to insert 1 ms of silence at the beginning of the file. This movie shows the steps.
For an acoustic measurement, this is called Time-of-Flight. While this isn’t an acoustic measurement our measurement system doesn’t know that (shhhhh!). While the TOF hasn’t changed the signal in any way, the phase response of the transfer function is no longer useful, since it includes the effect of the Time-of-Flight. There are now thousands of degrees of phase shift over the bandpass of the pulse. Watch the movie.
One way to understand what has happened is to consider that all frequencies are playing continuously. By introducing a 1 ms delay, there is now a frequency-dependent phase shift. Note that I placed the cursor at 1 kHz in the movie, and the corresponding phase shift was about 360 degrees. This makes perfect sense since the period of 1 kHz is 1 ms. We’ve have delayed the signal a complete cycle at 1 kHz. Since higher frequencies have shorter periods, they exhibit much more phase shift. The unwrapped phase response clearly shows this. It is common practice to “wrap” the phase response to make it more compact.
What can we take away from this? “Wrapping” in the phase response is the signature of delay in the transfer function. The delay must be removed to allow the phase response of the “system” to be observed.
Group Delay – A Different Perspective
That little 1 ms of delay seems to have wreaked havoc on the transfer function. One reason that it appears so is that phase is a very high resolution way of looking at time. It should only be used after we are “zeroed in” on the signal. A better initial view of the timing in the frequency domain is group delay, and a better name for group delay could be “frequency delay.” It shows the frequency-dependent delay in milliseconds instead of degrees of phase shift. Note that the group delay is perfectly flat at 1 ms, which is exactly how much delay we introduced in Wavosaur. Measurement systems provide multiple perspectives for observing signals. Something that is hidden or confusing from one perspective may be very clear from another. You will appreciate group delay when aligning the sections of a multi-way loudspeaker. Arta provides a dialog for compensating for this delay. Watch the movie.
Ta-Da! Now the Phase Makes Sense
With Time-of-Flight compensation, the phase response is the same as before the delay was introduced. What have we learned?
1. A perfect impulse in time contains all frequencies, and they are all in-phase. This is the “holy grail” of transfer functions. This is what some think they are getting when they buy “boutique” speaker cables.
2. Real-world systems introduce delay from a variety of sources. The biggies are digital latency and air path delay. Our 1 ms “simulated” TOF could be tens of milliseconds in a real-world system. The method shown for excess phase (TOF) compensation is the same – use group delay to get a “one number” correction factor in milliseconds. In my ideal example, 1 ms was correct for all frequencies. In real life it won’t be. We’ll deal with that later.
3. TOF must be compensated if the transfer function is to be valid. I showed how to use group delay for this purpose. There are other methods.
4. The frequency response magnitude was blind to all of this. It shows “what” gets there, where as phase and group delay show “when.” I will show some examples of when this matters in future installments.
5. The issues presented are a factor for all “time selective” measurement systems. If you fail to compensate for TOF when equalizing your system, the displayed response (and your equalizer settings!) can be completely bogus.
6. Practice make perfect. Lock yourself in a room with your computer and try some variations on what I have shown here. Then apply the principles to your measurement system of choice.
By doing this in a virtual environment, we are free of the distractions of the real-world – no cables, no mics, no loudspeakers, no screaming kids or passing trucks. You can practice on an airplane or in a hotel room, but not while driving.
In the next installment I will band-limit the Dirac pulse with high pass and low pass filters, making it a band-limited system. All real-world systems are bandpass filters and we need to be able to recognize how this affects the transfer function. This opens the door to investigating lots of interesting audio topics, such as analog vs. digital filtering techniques.
If you want more background on the terminology used in this blog, you need look no further than SynAudCon Level 100 – Principles of Audio training. pb