Most of the acoustic phenomena we observe as audience members or suffer as engineers are related to acoustic phase. Remember that in previous articles we saw that the operating principle of line arrays was based on phase, and as for subwoofer arrays, the phase relationship according to the distance between omnidirectional elements likewise determined the system's coverage and sound pressure.
Polarity or phase
But wasn't phase that button we find at the top of each channel on a mixing console? Or those gadgets we put in front of loudspeakers that tell us whether the speaker is fine or not, depending on whether the green or red light comes on?
Well, the truth is no, or at least it's not that simple, because what these little boxes, wrongly called "phase testers," actually tell us is whether the speaker's first excursion is forward (positive, green light) or backward (negative, red light), as a result of a reversed connection. In the case of the "phase inverse" button on a console channel, what we are doing is swapping the hot conductors, that is, pin 2 and pin 3 of the XLR input.
The impulse graph also shows time versus amplitude
In both cases, the correct names would be polarity tester and polarity inverter. Because, in reality, polarity inversion is what causes a 180º difference in the audio signal. We know that sound is measured in "Hertz," which is the unit of frequency: the more cycles per second, the higher the frequency.
There is a clear relationship between the difference in phase degrees and the resulting effect on total level
Breaking down a complete cycle, on the vertical plane we have the amplitude of the signal, which will also tell us its polarity, positive or negative half-cycle. On the horizontal plane we have time; remember that the time equivalent to one complete cycle is what we know as the period, and it is calculated with the following formula: T= 1/ F, that is, the period is the inverse of the frequency.
Applying this formula, the higher the frequency, the shorter the period, so for 10KHz the period will be 0.1ms, 1ms for 1KHz and 10ms for a low frequency of 100Hz. We also see that the horizontal plane shows the phase of the signal, which for a complete cycle goes from 0º to 360º, with 180º being half the cycle, that is, when the signal crosses level 0.
To see whether a signal has inverted polarity, that is, 180º, you need to measure the impulse response. You can measure a piece of electronic equipment to see whether the impulse is positive or negative, but you can also measure the acoustic impulse, that is, the signal produced by a loudspeaker or a complete sound reinforcement system.
- If we have two signals of the same level with a 0º difference (time difference 0) and sum them, we get a resulting signal 6dB higher.
- If we have two signals of the same level with a 45º difference and sum them, we get a resulting signal close to 6dB.
- If we have two signals of the same level with a 90º difference and sum them, we get a resulting signal of 3dB.
- If we have two signals of the same level with a 120º difference and sum them, we get a resulting signal of 0dB.
- If we have two signals of the same level with a 150º difference and sum them, we get a resulting signal of -6dB.
- If we have two signals of the same level with a 180º difference and sum them, we get a resulting signal equal to zero, that is, cancellation or maximum attenuation.
- If we kept summing signals, we would have: 210º = -6dB
- 240º = 0dB
- 270º =+3dB
- 322º =+5.5dB
- 360º =+6dB
How an analyzer displays it
Ideally, we would now use an analyzer to see when these phase differences occur and their consequences on the resulting signal, that is, the system's total frequency response. We would need to be able to display the phase relationship as a function of frequency, so we have to find a way to represent the trigonometric phase circle graphically on an analyzer screen. The phase graph is one of the analyzer windows that most sound engineers still find hard to interpret; we will now explain how to read it, which in reality is not that complex.
Today there are several software analyzers for PC and Mac, such as Smaart Live, Spectralab, Impulse, Mac Foh, Spectrafoo... but there are also specialized measurement systems, such as Meyer Sound's world-famous SIM3. The phase graph shows the delay of some frequencies relative to others; the speed of sound is the same for all frequencies and varies proportionally with temperature.
Although this speed is fixed (in our examples, 340m/s at 15ºC), frequencies are not reproduced at the same time, even within the same loudspeaker. This is due to the physical design of the cabinets, since the high-frequency driver is not aligned with the woofer, so not all frequencies come out at once. But it can also be due to poorly designed filters. Remember that, depending on the type of filter used, we will get delays, and these will be 45º per order. If we have several loudspeakers in a system, these time differences can become even greater.
A high frequency may be delayed relative to a mid frequency, and that one relative to low frequencies; the delay can range from 0º to several cycles of difference, not just up to 360º (1 cycle)
Almost no manufacturer includes phase graphs in the technical specifications of their user manuals. This would be like taking an acoustic X-ray of the cabinet. Another manufacturer we often mention in these articles makes all its loudspeakers phase-compatible, which means we can always combine any model without any problem.
On a phase graph, we would see that the vertical axis shows degrees, ranging not from 0º to 360º but from -500º to +500º; in other words, the delay of the highest frequency on the horizontal axis relative to the lowest, in this case, is 1000º, which is equivalent to two cycles and 280º. But imagine that instead of that delay we had a larger one, of 10 cycles, or 3600º; it would be quite hard to graph it in full, so the solution adopted in most software and analyzers is to plot everything within a single cycle so that, no matter how many cycles of delay there are between frequencies, we don't need an endless graph. This is possible. On such a graph we would see that the vertical straight lines do not count as signal, but as a wraparound transition in the graph, so the corresponding delay can continue to be plotted.
In another example we would see that there is one cycle of delay between 125Hz and 50Hz and one cycle of delay between roughly 800Hz and 125Hz. However, from 2KHz to 4KHz there is no delay and they are practically close to 0º. Finally, across the entire audio spectrum, between 50Hz and 16KHz, there is a phase variation of 1260º.
So far we have been looking at phase graphs and always talking about one signal, but if we had two signals, we would see the sum of the two; so ideally we would take one of the signals, store it in the analyzer's memory, and then measure the second signal on its own. By overlaying the signal we are measuring with the one stored in memory, depending on the existing phase differences, we will get a frequency response determined by those phase differences. So when both signals are active we will get summations and attenuations in the frequency response, and in the area where the phase difference between one signal and the other equals 180º there will be a cancellation or maximum attenuation.
We can also measure the phase response of a piece of electronic equipment. In the graph above, an equalizer has been measured with 1KHz boosted by 6dB; in the magnitude, the frequency response (bottom), we see how the phase response (top) has changed and now ranges from -30º to +30º. The signal used in the test for this example is pink noise.
To take phase response measurements, besides the analyzer, whether software or hardware, and the measurement microphone, we need to connect everything properly. To do this, we need to perform a transfer function, that is, compare a reference signal with a measurement signal.
This is because with a single microphone signal we could only perform spectrum or frequency analysis. RTA (real time analyzer)
The flaw in this type of measurement is that if a summation or cancellation occurs due to, for example, a reflection, the RTA will not be able to tell us that this is the cause; it will only show an increase or decrease in level. The sound engineer will correct this flaw by boosting or cutting the level in the affected area of the spectrum, but this is not the solution, since the problem is not one of frequency response but of time and, as a consequence, phase.
To measure the acoustic phase response of a sound system, we proceed in the same way: we have a measurement signal from the measurement microphone and a reference signal, pink noise, from the analyzer's generator, the console, a CD... which we send back via a cable or console split, since at the same time it goes into the analyzer it must go into the sound system so the comparison can be made.
Another detail we must not forget is the latency of the equipment used, such as loudspeaker processors, digital equalizers... and of the analyzer's own interface, since if there is latency there is delay, and if there is delay the phase response will not match what we hear.
Keep in mind that the signal coming out of our sound system must reach the analyzer at the same time as the signal going into the system, or in other words, the measurement and reference signals must arrive together for the transfer function comparison to be carried out correctly.
Practical applications
Putting this into practice means, for example, that if we have two loudspeakers and stand at the same distance from each, there is no time difference, so the resulting sound will be 6dB louder than if we were listening to a single loudspeaker; if we look at the circle, this holds true, since we have a 0º difference.
Remember that this 6 dB increase results from applying the formula 20log 2/1 when the sound source is doubled. Of course, this happens with all audio signals, since the distance will always be the same, but what happens if, for example, one cabinet is 5m away from us and the other 6.7m? Quite simply, the frequency for which the difference in distance is half its wavelength will be strongly attenuated.
In the example mentioned, D1-D2 = 6.7m - 5m = 1.7m, which means that for a frequency of 100Hz we will have strong attenuation or cancellation at the spot where we are standing, since, remember, wavelength comes from the formula V/F, where V is the speed of sound (in this case at 15ºC) and F the frequency, which gives us 340m/s / 100Hz = 3.4m. We said this would be 360º of that frequency, so 1.7m would correspond to 180º of 100Hz and, as explained above, the graphs will show maximum attenuation or cancellation.
From the theory applied, we can already sense that there will always be level differences at certain frequencies depending on where we are listening, so it will never be the same for the whole audience.
Clearly, the problem can't be solved completely, but it can be minimized so that the phase differences between different points in the audience are as small as possible. To that end, we should aim for a phase variation close to 0º at the FOH position so that, as we move from front to back and from right to left, those phase variations are not exaggerated and stay within acceptable margins.
Remember that between 0º and 120º we will have a 6dB difference, so it is not that hard to achieve. How to do it will be covered in upcoming chapters!
