# Science and Math for Audio Humans – Phase

• by Danny Maland
• Maestro, would you please play “The Disclaimer?”
• Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.
• As this series has progressed, the topic of phase has been nagging at the back of my mind. It's far more fundamental than some of the concepts that have been presented in the last little while, yet the “natural progression” of the topics has led away from a discussion of phase. Perhaps, like a sine wave, it's time for us to cycle back.
• So – what is phase, anyway? Phase is the measure of the time arrival difference between two pressure waves or signals of the same frequency, as experienced by a particular observer. It is usually expressed in degrees, although some folks might use radians. (For most of us, who are used to degrees, radians are a curious unit. Radians are found by dividing the length of an arc by the radius of that arc. One full rotation, then, is 2π radians. The ratio is 2πr/r, and the two “r” numbers – the radius – cancel out. Now you know!)
• The way this degree assignment works for a wave is illustrated in Figure 1. The beginning of the wave cycle is 0°, the “positive” peak is 90°, the zero-crossing is 180°, the “negative” peak is 270°, and the end of the cycle is 360°.

This being laid out, let's say that we have two 1kHz sine waves. Everything about them is the same, except that one of them arrive at our measurement point a half-millisecond behind the other. Since a 1kHz sine wave makes 1000 cycles every second, then 1ms is a single cycle. Half a millisecond, then, is one half of a cycle of a 1kHz wave. The half-cycle is where the zero-crossing occurs, so the “late” wave can be said to be 180° out of phase. Figure 2 shows this situation. The “late” wave begins its positive-going half-cycle where its counterpart is beginning its negative-going half-cycle.

Where things really become interesting is when our two waves are combined. For instance, let's say that this “half millisecond late” condition occurs at the diaphragm of a microphone. The microphone, unlike a human, has no brain to interpret what it experiences – and further, it is only a single transducer. We humans benefit from having two, spatially separated transducers attached to our brains, so we can make differential comparisons. A microphone, on the other hand, “reports” only the total pressure occurring at the diaphragm, so...

The microphone hears silence, because the sum of the two pressure waves is 0 (in terms of a differential from whatever starting point we have).

In other words, yes, sound pressure waves and signals do interfere. That interference can be constructive or destructive. If both waves arrive precisely in phase, they will add in a constructive manner. Please note that the “sum” wave at the bottom of Figure 4 has a peak amplitude of “2,” which is double the pressure of each individual wave. (+6 dB SPL, weighting and averaging time being inconsequential.)

At 90°, the summation looks like this.

The tools at wolframalpha.com were very handy for making these figures.

A few paragraphs ago, I set up the “180° out” example by being very specific. The two waves involved had a frequency of 1kHz, and the wave arriving “late” was 0.5ms behind the first wave. The reason I had to be so specific was because of this property of phase: For any given time arrival differential, different frequencies will be more or less out of phase. For instance, if the time differential was still 0.5ms, but the wave frequency was 2kHz, the phase angle would be 360°. If a 1kHz wave cycles once per millisecond, then a 2kHz wave has to cycle twice per millisecond. That being the case, at half a millisecond, the 2kHz wave has finished a full cycle (360°). By extension, a 3kHz wave completes one and a half cycles at 0.5ms, and so is 540° out of phase – which is effectively 180°. (Unless you need to be exact about the time differential, you can “start over” when going beyond 360°.)

In the vein of “interesting applications,” this “wrapping phase” phenomenon is what causes the effect known as “comb filtering.” This name comes from the characteristic look of the effect when viewed as an output trace from a FFT (Fast Fourier Transform) analyzer. In this case, the analyzer is ReaFIR, a rather nifty freebie from Cockos that you can read more about here.

Comb filtering is only apparent with broadband sounds, because the first complete null does not occur at any lower frequency than that which is 180° out of phase for a given time arrival differential. After that, a complete null will appear at every frequency where the time differential corresponds to a wave being effectively 180° out of phase. Interestingly, this reoccurance works out to be the first null frequency, plus the frequency of an octave above the first null. A first null at 1kHz will have additional nulls at 3kHz (1kHz + 2kHz), 5kHz, 7kHz, and so on. A first null at 2kHz will have additional nulls at 6kHz, 10kHz, 14kHz, and so on.

In the vein of bedrock concepts, the “wrapping phase” phenomenon is even more important – especially as it deals with a misapplication of terminology that is still quite rampant in the world of audio: “Phase Reverse” or “Phase Flip“

It is not my intention that you should become a pedantic lecturer or all those around you, but, please be aware:

There is no such thing as swapping, reversing, or flipping phase.

There is such a thing as swapping, reversing, or flipping POLARITY.

Because the phase of a frequency is dependent on a specific time arrival difference for that frequency, the only way to get the entire audible spectrum out of phase at once would be this: Have an infinitely large number of crossover filters (hey, 1kHz and 1.001kHz are not the same frequency), all able to selectively filter all but an infinitely small passband, which would then feed an infinite number of delay lines, each line tuned to produce exactly 180° of phase shift for its respective frequency, which would then feed a summing amplifier with an infinite number of inputs.

Not. Possible. In. This. Universe.

However, what is possible is to re-configure a transducer or signal path such that, in relationship to other devices, positive pressure corresponds to negative signal or vice-versa. In a correctly implemented system, this produces no time differential at all between two signals. There is an “appearance” of all frequencies being 180° out of phase, to be sure, but that is only an appearance. Again, there is no time differential involved, and so phase is unaffected.

(This, by the way, is why it is impossible to actually fix a phase problem by way of polarity inversion on selected signals. You may, of course, find that polarity flipping makes a phase problem less apparent, but to really solve the issue you have to do something that effects time.)