# Phase or Polarity?

The use of the term ‘flipping the phase’ or ‘inverting the phase’ is a common misnomer that I see time and time again by professional and amateur alike. The problem is that to invert the phase of an electrical signal simply does not make sense, and what is most commonly meant is actually a polarity inversion. But what’s the difference and why is the terminology wrong? To explain this, it is necessary to clarify what each of these terms mean.

Polarity
Polarity is a very simple concept. Any electrical signal has polarity, and it is a reference to the signal’s position or voltage above or below the median line. A device which inverts the polarity of a signal will simply swap positive voltage to negative voltage and vice versa. Electrically, it is as simple as reversing the positive and negative terminals. Electrically summing a signal with an inverted-polarity time-synchronous copy of itself results in complete cancellation.

Figure 1 shows an example of a polarity reversal on a simple sine wave. The red signal shows the input while the blue signal shows the output with inverted polarity. This is the result of pressing the ‘∅’ or ‘phase’ [sic] button on a mixing console / DSP, or wiring-up an XLR with pins 2 and 3 reversed. This process has no effect on the time of a signal.

Phase
The concept of phase is considerably more complex.

Phase is delay. It is also a relative concept and can be visualised as circular. Comprehensive school mathematics tells us that a circle has 360° of rotation before returning to the origin. The phase of a signal, relative to the starting point, is expressed in degrees. A 90° phase shift, as in a circle, is a quarter-rotation or a quarter of a wavelength. It follows that a 180° phase shift is a half-rotation or half-wavelength and a 360° phase shift is a whole rotation or full wavelength. This increase will continue infinitely; a 720° phase shift is 2 full wavelengths (or 2 rotations) and a 1040° phase shift is 3 wavelengths (3 rotations) etc.

Figure 2 shows a sine wave at 1kHz (in red). The green trace is 90° phase-shifted relative to the red trace. It starts a quarter of a wavelength (λ/4) later in time. Similarly, the blue trace is 180° phase-shifted (half-wavelength or λ/2) relative to red and the brown trace is 270° phase-shifted (three-quarter-wavelength or 3λ/4) relative to red. A 360° phase shift is a full-cycle, and would overlay the red trace.

As you can see, the starting points of the 4 traces are staggered by 0.25ms, starting with 0° (red), then 90° (green), 180° (blue) and 270° (brown). It follows that a 90° phase shift is equal to a 0.25ms delay, a 180° phase shift is equal to a 0.5ms delay and a 270° phase shift is equal to a 0.75ms delay, but as phase is wavelength (and hence frequency) dependent, these figures are correct at 1kHz only.

Phase can be expressed as a positive or negative amount. In a cyclic signal, such as a sine wave, a 180° phase shift is also equal to a -180° phase shift, so 270° is equal to a -90° phase shift and 360° is equal to a 0° phase shift. This holds true indefinitely, as 0° is also equal to 360°, 720°, 1080°, 1440° etc. It’s easy to see why phase is often ‘wrapped’ like this (between -180° and 180°) for clarity.

What’s the difference?

You can see by comparing figure 2 to figure 1 that a 180° phase shift appears to be equivalent to a polarity reversal, but this is most definitely not the case. Remember that phase is delay, and the 180° phase shift was accomplished by delaying the signal by 0.5ms, whereas the polarity reversal was achieved by swapping the positive and negative voltages. The polarity reversal has no delay.

A sine wave at 2kHz has a period of 0.5ms, so a 0.5ms delay would create a 360° phase shift and a 0.25ms delay would create a 180° phase shift.

A dual-sine signal consisting of 1kHz and 2kHz sine waves would react differently to delay with regards to phase. If we apply a 0.5ms delay to this signal, it would result in a 360° phase shift at 2kHz but only a 180° shift at 1kHz.

Figure 3 shows a 1kHz + 2kHz dual-sine wave in red, with a polarity inversion in blue. Figure 4 shows the same 1kHz + 2kHz dual-sine wave in red, but with a 0.5ms delay applied, equivalent to a 180° phase shift at 1kHz and a 360° phase shift at 2kHz.

Applying a delay to a complex signal (such as pink noise or music) and comparing it to the original results in a phase shift that varies over frequency. As the wavelengths at low frequencies are long, the relative delay with respect to the period would also be low compared to the same delay at high frequencies, as will the phase. For a constant delay, as the frequency increases, the phase shift will also increase.

Figure 5 shows a phase response chart, with phase on the y-axis and frequency on the x-axis. Also shown is a magnitude response. This graph shows a 1ms delay applied to a full range signal and referenced to the original. We can see the phase shift increasing rapidly.

• At 100Hz, the period is 10ms, the phase shift is 36°
• At 500Hz, the period is 5ms, the phase shift is 180°
• At 1kHz, the period is 1ms, the phase shift is 360°
• At 5kHz, the period is 0.2ms, the phase shift is 1800°
• At 10kHz, the period is 0.1ms, the phase shift is 3600°

The phase shift appears to speed up, but this is because it is plotted on a logarithmic chart, as are the majority of audio frequency charts. For a given delay, phase increases linearly with frequency, which makes it appear to be an exponential rise on a logarithmic graph.

Figure 5 also shows the concept of ‘wrapping’. It would be impossible and counterintuitive to display a phase trace from 0° all the way up to 7200° (the maximum phase shift on this particular graph), and that’s with just a 1ms delay! Greater delays would result in more and more phase shift.

Conclusion
Both polarity and phase are fundamental in the art of sound system design, and an understanding of the concepts are rudimentary to this. Hopefully you can see that phase and polarity are not interchangeable. I believe that there is one main reason why these two concepts are so often confused, and that is that professional console manufacturers so often refer to the polarity reversal function on their console as ‘reversing the phase’ or ‘activating a 180 degree phase change’ (the notable exception being Allen & Heath). Of all people, they should know better!

It may also have something to do with a 180° phase shift at a single cyclic frequency appearing identical to a polarity reversal in the time domain!

While it is often straightforward to distinguish between the two concepts regardless of language used, I believe it is important to strive for accuracy as laziness with terminology can commonly cause confusion. The next step is for manufacturers to change their ways!

Tags: , ,