Phase relationships in audio signals

Phase is probably most easily understood in terms of one of the fundamental building-blocks of audio: the sine wave.

A sine wave - described, in audio terms, as a pure tone - is a repeating cycle of regular frequency and amplitude. The shape of the wave can be plotted from the sines (hence ‘sine wave’) of any series of numbers following an arithmetic progression.

Here, the sine of each number in an arithmetic progression is plotted on the Y-axis against the arithmetic progression on the X-axis.

This shows a single cycle.

A sine wave can also be plotted against time from a single point on a circle revolving at a constant speed.

Here, a point on a circle is plotted on the Y-axis against degrees of rotation (which correspond with phase angle) on the X-axis.

In the diagram, a point on a rotating circle describes a curve that rises from zero (0°) to maximum positive value (90°) before returning to zero (180°) and continuing on the negative half of the cycle. A full cycle is completed at 360°.

Sine waves often arise from rotation: the shape of a standard UK 230V AC waveform, generated by a coil rotating with respect to a magnetic field at a constant speed (50 Hz), is a sine wave.

In air (the medium through which we usually experience sound) the positive half of the waveform represents compression of air molecules, and the negative half represents rarefaction.

Wavelength, usually represented by the Greek letter lambda (λ), is the distance sound travels during a single cycle. This can be calculated by dividing the speed of sound (343 m/s at 20°C at sea-level) by the frequency (cycles/second, expressed in Hertz). For example, the wavelength of a 1 kHz sine wave is 343/1,000 metres, or approximately 34 cm.

From this we can derive a relationship between phase and wavelength:

In this diagram, showing two complete cycles, we can see that a quarter of a wavelength (λ/4) corresponds with a phase angle of 90°, and half a wavelength corresponds with a phase angle of 180°.

Where a single sine wave is produced by two separate loudspeakers, any difference in their distance from the listening/measuring position can be described in fractions of a wavelength (and, in turn, phase difference).

Here, two loudspeakers are at different distances from the listening position. For any given frequency the difference in distance can be expressed as a fraction of wavelength (d2 − d1 = λ/x). The difference in distance (d2 − d1) will therefore affect the phase relationship between them, and (see below) the resultant sound.

Some of the effects of such phase differences on measured output are shown in the graphs below:

Here, the loudspeakers are equidistant from the measuring position (i.e. the two signals are in phase). The reason you cannot see the ‘original’ sine wave on the graph is that it is masked by the second identical (0°) signal.

The result is a doubling of sound pressure (the equivalent of a quadrupling of power): the resulting level is 6 dB greater than that of a single loudspeaker.

Here, the loudspeakers differ in distance from the measuring position by a quarter of a wavelength (90°). We can see that the combined signal, although differing in magnitude and phase from either of the source signals, is itself a sine wave of the same frequency as its ‘parents’.

The amplitude of the combined signals is greater than the output from a single loudspeaker. However, it is now only 1.414 (√2) times - 3 dB - greater.

Note that 3 dB is the nominal increase assumed where complex signals from two separate sources are equal in level but unrelated in phase: to calculate the total acoustic output of a system with multiple loudspeakers that are not coupled, you would add 3 dB to the output of a single loudspeaker for each doubling of number (i.e. the output of two loudspeakers would be 3 dB greater than one; the output of four loudspeakers would be 3 dB greater than two and 6 dB greater than one; the output of eight loudspeakers would be 3 dB greater than four and 9 dB greater than one; etc.).

Here, the loudspeakers differ in distance from the measuring position by a third of a wavelength (120°).

Now, the sum is equal to the output from a single loudspeaker: in this position, the second loudspeaker has no effect on the overall level.

Where the loudspeakers differ in distance from the measuring position by 5/12 of a wavelength (150°), the sum is less than the output of a single loudspeaker: in this position, disconnecting the further loudspeaker would result in an increase in level at that frequency.

Finally, where the loudspeakers differ in distance from the measuring position by half a wavelength (180°), we can see that the two sounds cancel each other completely.

In practice, we never experience sound in a single position (most of us have two ears), and in any indoor event reflected as well as direct sound will reach any listening position, so we would experience a very substantial reduction in level rather than total silence.

This has implications for sound systems employing multiple loudspeakers. A key distance between driver centres in arrayed loudspeaker systems is half a wavelength (λ/2):

In this diagram, two loudspeakers are separated by a distance of half a wavelength of the measured frequency. We can see that directly on axis, the measuring position is the same distance from each loudspeaker. As a result, the sum of their outputs in that position is 6 dB greater than the output of a single loudspeaker.

Off-axis, the loudspeakers are no longer equidistant, but as long as the distances differ by less than λ/3 there is still an increase in level.

Moving further off axis, however, we reach a point where the difference is equal to λ/3. Beyond this point, the combined level becomes less than that of an individual loudspeaker until, at the sides, the difference in distance is λ/2, and the outputs cancel.

We can see here that arraying loudspeakers in this way can be used to control output pattern. We can also see that the difference in level between an on-axis position and the first off-axis position is 3 dB: if an array like this can be angled so that nearer audience positions are at a 90° phase angle, whilst further audience positions are at 0°, we can reduce the level-drop over distance to something approaching 3 dB. This is one aspect of how line-arrays work.

We can see that interference between the two loudspeakers is constructive (it increases the overall level) through an arc of forward positions. As we move outside that arc, interference becomes destructive (it reduces the overall level).

The output pattern of an arrayed system can also be changed using delay:

Here, the output of one of the loudspeakers has been delayed (so that its effective position is further away, as shown by the dotted lines). This moves the coupled position away from the centre. This method can be used to angle the effective output of the array.

Varying the delay and level of individual drivers in a straight array to control output pattern was pioneered by Airo in installations (primarily in houses of worship) in the 1960s. It allows the output of a column of loudspeakers mounted on a wall or pillar to project downwards without angling the column away from the wall.

Delaying the outermost cabinets in an array widens the outward spread of sound without altering the displacement of the cabinets.

All the examples above consider the effects of phase at a single frequency. In practice, any loudspeaker system produces sound over a range of frequencies, and any difference in time (delay) or distance does not act equally at all frequencies.

For example, two loudspeakers that differ in distance from the measuring position by 17 cm will be half a wavelength (180°) apart at 1,000 Hz. However, they will be a whole wavelength apart (i.e. effectively 0°) at 500 Hz , 1.5 wavelengths apart (effectively 180°) at 750 Hz, and so on, resulting in alternating cancellation and reinforcement at higher frequencies. The result (shown in the graph below) is known as comb-filtering, because the graph - especially where frequencies are plotted as an arithmetic rather than a logarithmic progression - looks like the teeth of a comb:

We can see that the effect is limited to frequencies above that at which separation = λ/2. We can also see from the two previous diagrams that where the loudspeakers are separated by half a wavelength or less, complete cancellation does not occur in forward positions.

If we can get the loudspeaker centres closer than half a wavelength of the highest frequency in their range, therefore, we will have achieved effective coupling without comb-filtering through an arc of forward positions, together with destructive interference in positions that lie further off-axis. This is another aspect of how line-arrays work.

You should note that although the discussion above addresses loudspeakers, phase relationships affect multiple microphones (or multiple pickups) on a single sound source in exactly the same way. For that reason, it is generally good practice to avoid using multiple pickup methods on a single sound source.

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