Time shifts

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:

Canons: combining two copies of a signal with a time shift sufficient for the signal to change appreciably, we might hear the two as separate musical streams, in effect comparing the signal to its earlier self. If the signal is a melody, the time shift might be comparable to the length of one or several notes.

Echos: At time shifts between about 30 milliseconds and about a second, the later copy of the signal can sound like an echo of the first one. An echo may reduce the intelligibility of the signal (especially if it consists of speech), but usually won't change the overall ``shape" of melodies or phrases.

Filtering: At time shifts below about 30 milliseconds, the copies are too close together in time to be perceived separately, and the dominant effect is that some frequencies are enhanced and others suppressed. This changes the spectral envelope of the sound.

Altered room quality: If the second copy is altered to be much quieter than the first, and moreover if we add a third, or even a thousand more delayed copies at reduced amplitudes, they can mimic the echos that arise in a room or other acoustic space.

The overall quality of a given arrangement of delayed copies of a signal may combine two or more of these affects.

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.