otherwise they are out of phase with each other. In the clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise. The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by a microphone. This is usually the case in linear systems, when the superposition principle holds. For arguments when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that constructive interference is occurring. At arguments when the phases are different, the value of the sum depends on the waveform. For sinusoids For sinusoidal signals, when the phase difference is 180° ( radians), one says that the phases are opposite, and that the signals are in antiphase. Then the signals have opposite signs, and destructive interference occurs. Conversely, a phase reversal or phase inversion implies a 180-degree phase shift. When the phase difference is a quarter of turn (a right angle, or ), sinusoidal signals are sometimes said to be in quadrature (e.g., in-phase and quadrature components). If the frequencies are different, the phase difference increases linearly with the argument . The periodic changes from reinforcement and opposition cause a phenomenon called beating. For shifted signals The phase difference is especially important when comparing a periodic signal with a shifted and possibly scaled version of it. That is, suppose that for some constants and all . Suppose also that the origin for computing the phase of has been shifted too. In that case, the phase difference is a constant (independent of ), called the 'phase shift' or 'phase offset' of relative to . In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant. In this case, the phase shift is simply the argument shift , expressed as a fraction of the common period (in terms of the modulo operation) of the two signals and then scaled to a full turn: If is a "canonical" representative for a class of signals, like is for all sinusoidal signals, then the phase shift called simply the initial phase of . Therefore, when