Phase

FREQUENCY SPECTRA OF NON-REPETITIVE SOUNDS

Non -repetitive waveforms do not have a recognizable pitch and sound noise-like. Their frequency spectra are likely to consist of a collection of components at unrelated frequencies, although some frequencies may be more dominant than others. The analysis of such waves to show their frequency spectra is more complicated than with repetitive waves but is still possible using a mathematical technique called Fourier transformation, the result of which is a frequency-domain plot of a time-domain waveform.

Single, short pulses can be shown to have continuous frequency spectra which extend over quite a wide frequency range, and the shorter the pulse the wider its frequency spectrum but usually the lower its total energy.

Random waveforms will tend to sound like hiss, and a completely random waveform in which the frequency, amplitude and phase of components are equally probable and constantly varying is called white noise.

A white noise signal’s spectrum is flat, when averaged over a period of time, right across the audio-frequency range (and theoretically above it). White noise has equal energy for a given bandwidth, whereas another type of noise, known as pink noise, has equal energy per octave. For this reason, white noise sounds subjectively to have more high-frequency energy than pink noise.

PHASE

Two waves of the same frequency are said to be ‘in phase’ when their compression (positive) and rarefaction (negative) half-cycles coincide exactly in time and space.

1. If two in-phase signals of equal amplitude are added together, or superimposed, they will sum to produce another signal of the same frequency but twice the amplitude.

2. Signals are said to be out of phase when the positive half-cycle of one coincides with the negative half-cycle of the other. If these two signals are added together, they will cancel each other out, and the result will be no signal.

3. Clearly, these are two extreme cases, and it is entirely possible to superimpose two sounds of the same frequency which are only partially in phase with each other. The resultant wave in this case will be a partial addition or partial cancelation, and the phase of the resulting wave will lie somewhere between that of the two components.

Phase differences between signals can be the result of time delays between them. If two identical signals start out at sources equidistant from a listener at the same time as each other then they will be in phase by the time they arrive at the listener. If one source is more distant than the other then it will be delayed, and the phase relationship between the two will depend upon the amount of delay.

A useful rule-of-thumb is that sound travels about 30 cm (1 foot) per millisecond, so if the second source in the above example were 1 meter (just over 3 ft) more distant than the first it would be delayed by just over 3 ms. The resulting phase relationship between the two signals, it may be appreciated, would depend on the frequency of the sound, since at a frequency of around 330 Hz the 3 ms delay would correspond to one wavelength and thus the delayed signal would be in phase with the undelayed signal. If the delay had been half this (1.5ms) then the two signals would have been out of phase at 330Hz.

Phase is often quoted as a number of degrees relative to some reference, and this must be related back to the nature of a sine wave.

The height of the spot varies sinusoidally with the angle of rotation of the wheel. The phase angle of a sine wave can be understood in terms of the number of degrees of rotation of the wheel.

A sine wave may be considered as a graph of the vertical position of a rotating spot on the outer rim of a disc (the amplitude of the wave), plotted against time. The height of the spot rises and falls regularly as the circle rotates at a constant speed. The sine wave is so called because the spot’s height is directly proportional to the mathematical sine of the angle of rotation of the disc, with zero degrees occurring at the origin of the graph and at the point shown on the disc’s rotation in the diagram. The vertical amplitude scale on the graph goes from minus one (maximum negative amplitude) to plus one (maximum positive amplitude), passing through zero at the halfway point. At 90 °of rotation the amplitude of the sine wave is maximum positive (the sine of 90 °is 1), and at 180 °it is zero (sin 180 ° = 0). At 270 °it is maximum negative (sin 270 ° =-1), and at 360 °it is zero again. Thus, in one cycle of the sine wave the circle has passed through 360 ° of rotation.

*The lower wave is 90 °out of phase with the upper wave.

It is now possible to go back to the phase relationship between two waves of the same frequency. If each cycle is considered as corresponding to 360 °, then one can say just how many degrees one wave is ahead of or behind another by comparing the 0 °point on one wave with the 0 °point on the other. In the example wave 1 is 90 °out of phase with wave 2. It is important to realize that phase is only a relevant concept in the case of continuous repetitive waveforms and has little meaning in the case of impulsive or transient sounds where time difference is the more relevant quantity. It can be deduced from the foregoing discussion that (a) the higher the frequency, the greater the phase difference which would result from a given time delay between two signals, and (b) it is possible for there to be more than 360 °of phase difference between two signals if the delay is great enough to delay the second signal by more than one cycle. In the latter case it becomes difficult to tell how many cycles of delay have elapsed unless a discontinuity arises in the signal, since a phase difference of 360 ° is indistinguishable from a phase difference of 0 °.

Bibliography:

1. Sound and Recording, Sixth Edition, Francis Rumsey and Tim McCormick.

2. Designing Sound, MIT

3. Sound Design, Maurizio Giri.