User Guide

Sound System Design Reference Manual
Combining Sine Waves
Referring to Figure 1-2, if two or more sine
wave signals having the same frequency and
amplitude are added, we find that the resulting signal
also has the same frequency and that its amplitude
depends upon the phase relationship of the original
signals. If there is a phase difference of 120°, the
resultant has exactly the same amplitude as either
of the original signals. If they are combined in phase,
the resulting signal has twice the amplitude of either
original. For phase differences between l20° and
240°, the resultant signal always has an amplitude
less than that of either of the original signals. If the
two signals are exactly 180° out of phase, there will
be total cancellation.
In electrical circuits it is difficult to maintain
identical phase relationships between all of the sine
components of more complex signals, except for the
special cases where the signals are combined with
a 0° or 180° phase relationship. Circuits which
maintain some specific phase relationship (45°, for
example) over a wide range of frequencies are fairly
complex. Such wide range, all-pass phase-shifting
networks are used in acoustical signal processing.
When dealing with complex signals such as
music or speech, one must understand the concept
of coherence . Suppose we feed an electrical signal
through a high quality amplifier. Apart from very small
amounts of distortion, the output signal is an exact
replica of the input signal, except for its amplitude.
The two signals, although not identical, are said to
be highly coherent. If the signal is passed through a
poor amplifier, we can expect substantial differences
between input and output, and coherence will not be
as great. If we compare totally different signals, any
similarities occur purely at random, and the two are
said to be non-coherent.
When two non-coherent signals are added, the
rms (root mean square) value of the resulting signal
can be calculated by adding the relative powers of
the two signals rather than their voltages. For
example, if we combine the outputs of two separate
noise generators, each producing an rms output of
1 volt, the resulting signal measures 1.414 volts rms,
as shown in Figure 1-3.
Figure 1-3. Combining two random noise generators
1-2
Figure 1-2. Vector addition of two sine waves