## Multiplying signals

In the previous tutorial we added sine tones together to make a complex
tone. In this chapter we will see how a very different effect can be
achieved by *multiplying* signals. Multiplying one wave by
another - i.e., multiplying their instantaneous amplitudes, sample
by sample - creates an effect known as *ring modulation* (or,
more generally, *amplitude modulation*). ‘Modulation’ in this
case simply means change; the amplitude of one waveform is changed
continuously by the amplitude of another.

In our example patch, we multiply two sinusoidal tones. Ring modulation (multiplication) can be performed with any signals, and in fact the most sonically interesting uses of ring modulation involve complex tones. However, we'll stick to sine tones in this example for the sake of simplicity, to allow you to hear clearly the effects of signal multiplication.

The tutorial patch contains two cycle~ objects, and the outlet of each one is connected to one of the inlets of a *~ object. However, the output of one of the cycle~ objects is first scaled by an additional *~ object, which provides control of the over-all amplitude of the result. (Without this, the over-all amplitude of the product of the two cycle~ objects would always be )

## Tremolo

When you first open the tutorial patch, a loadbang object
initializes the frequency and amplitude of the oscillators. One
oscillator is at an audio frequency of Hz. The other
is at a sub-audio frequency of Hz (one cycle every
ten seconds). The 1000 Hz tone is the one we hear (this is
termed the *carrier* oscillator), and it is modulated by
the other wave (called the *modulator*) such that we hear
the amplitude of the 1000 Hz tone dip to 0 whenever the 0.1 Hz
cosine goes to 0. (Twice per cycle, meaning once every five
seconds.)

The amplitude is equal to the product of the two waves. Since the peak amplitude of the carrier is 1, the over-all amplitude is equal to the amplitude of the modulator.

With the modulator rate set at *tremolo*.
(Note that this is distinct from *vibrato*, a term
usually used to describe a periodic fluctuation in pitch
or frequency.) The perceived rate of tremolo is equal to
two times the modulator rate, since the amplitude goes
to 0 twice per cycle. As described on the previous page,
ring modulation produces the sum and difference frequencies,
so you're actually hearing the frequencies 1001 Hz
and 999 Hz, and the 2 Hz beating due to the interference
between those two frequencies.

In these cases the rate of tremolo borders on the audio
range. We can no longer hear the tremolo as distinct
fluctuations, and the tremolo just adds a unique sort
of ‘roughness’ to the sound. The sum and difference
frequencies are now far enough apart that they no longer
fuse together in our perception as a single tone, but they
still lie within what psychoacousticians call the critical
band. Within this *critical band* we have trouble hearing
the two separate tones as a pitch interval, presumably
because they both affect the same region of our basilar
membrane.

## Sidebands

At a modulation rate of 32 Hz, you can hear the two tones as a pitch interval (approximately a minor second), but the sensation of roughness persists. With a modulation rate of 50 Hz, the sum and difference frequencies are 1050 Hz and 950 Hz - a pitch interval almost as great as a major second - and the roughness is mostly gone. You might also hear the tremolo rate itself, as a tone at 100 Hz.

You can see that this type of modulation produces new frequencies not present in the carrier and modulator tones. These additional frequencies, on either side of the carrier frequency, are often called sidebands.

At certain modulation rates, all the sidebands are aligned in a harmonic relationship. With a modulation rate of 200 Hz, for example, the tremolo rate is 400 Hz and the sum and difference frequencies are 800 Hz and 1200 Hz. Similarly, with a modulation rate of 500 Hz, the tremolo rate is 1000 Hz and the sum and difference frequencies are 500 Hz and 1500 Hz. In these cases, the sidebands fuse together more tightly as a single complex tone.

**Technical detail:**Multiplication of waveforms in the time domain is equivalent to

*convolution*of waveforms in the frequency domain. One way to understand convolution is as the superimposition of one spectrum on every frequency of another spectrum. Given two spectra

*S1*and

*S2*, each of which contains many different frequencies all at different amplitudes, make a copy of

*S1*at the location of every frequency in

*S2*, with each copy scaled by the amplitude of that particular frequency of

*S2*.

Since a cosine wave has equal amplitude at both positive and negative frequencies, its spectrum contains energy (equally divided) at both

*f*and

*-f*. When convolved with another cosine wave, then, a scaled copy of (both the positive and negative frequency components of) the one wave is centered around both the positive and negative frequency components of the other.

## Summary

Multiplication of two digital signals is comparable
to the analog audio technique known as *ring modulation*.
Ring modulation is a type of* amplitude modulation* -
changing the amplitude of one tone (termed the *carrier*)
with the amplitude of another tone (called the *modulator*).
Multiplication of signals in the time domain is equivalent to
convolution of spectra in the frequency domain.

Multiplying an audio signal by a sub-audio signal results
in regular fluctuations of amplitude known as *tremolo*.
Multiplication of signals creates *sidebands* - additional
frequencies not present in the original tones. Multiplying two
sinusoidal tones produces energy at the sum and difference
of the two frequencies. This can create beating due to
interference of waves with similar frequencies, or can
create a fused complex tone when the frequencies are harmonically
related. When two signals are multiplied, the output amplitude
is determined by the product of the carrier and modulator amplitudes.