Next: Complex numbers
Time shifts and delays
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 ideally should sound the same.
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. The difference, whatever it is,
clearly resides in neither of the two individual sounds, but rather in the
between the two. This interference can be perceived in at least four different
The sound of a given arrangement of delayed copies of a signal may
combine two or more of these affects.
- 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
- 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 earlier 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
- 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 played more quietly than the
first, and especially if we add many more delayed copies at reduced amplitudes,
the result can mimic the echos that arise in a room or other acoustic space.
Mathematically, the effect of a time shift 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 analyses, but in this and later
chapters the formulas will become more complicated and we will need more
powerful mathematical tools to manage them. In a preliminary
section of this chapter we will develop the additional background needed.
Next: Complex numbers