At 5:00 some afternoon, put on your favorite recording of the Ramones' string quarter number 5. The next Saturday, play the same recording at 5:00:01, one second later in the day. The two playings should sound the same, with any differences more likely to be of context or environment than to be intrinsic to the sound. Shifting the whole thing one second (or, if you like, a few days and a second) has no physical effect on the sound.
But now suppose you played it at 5:00 and 5:00:01 on the same day (on two different playback systems, since the music lasts much longer than one second.) Now the sound is much different. Moreover, the difference, whatever it is, clearly resides in neither of the two individual sounds, but rather in the interference between the two. This interference can be perceived in at least four different ways:
Mathematically, the effect of a time shoft on a signal can be described as a phase change of each of the signal's sinusoidal components. The phase shift of each component is different depending on its frequency (as well as on the amount of time shift). In the rest of this chapter we will often consider superpositions of sinusoids at different phases. Heretofore we have been content to use real-valued sinusoids in our analysis, but in this and later chapters the formulas will become more complicated and we will need more powerful mathematical tools to manage them. So in a preliminary section of this chapter we will develop the additional background needed.