Sound Synthesis Theory/Modulation Synthesis
Modulation Synthesis 
When we talk about modulation from an audio synthesis point of view, we refer to a time-varying signal (the carrier) being affected in some way by another (the modulator). Modulation can be found in a range of different sound effects and synthesis techniques and some of these effects occur naturally and help us to identify certain types of sound; for instance the commonly found performance styles of tremolo (modulation of amplitude) and vibrato (modulation of frequency) that are used in many stringed instruments are examples of this. Modulation is typical in synthesis because it enriches the character of the sound and also adds to the variance in timbre / character over time which is so often found in nature.
In the two basic methods of modulation synthesis that occur, ring modulation and amplitude modulation, there are two unique types of signal that occur in each method: bipolar and unipolar signals. A bipolar signal is the type of signal we have been examining in previous chapters, it has both a negative and positive amplitude and the waveform generally "rests" around zero in a time-domain plot. A unipolar signal is a bipolar signal that has been constant-shifted, that is, a constant value added to the overall signal to shift it into a range above zero, typically between 0 and 1. The reason for these two different types of signal follows.
Ring Modulation 
Ring modulation is the multiplication of two bipolar audio signals by each other. Each value of a carrier signal, C, is multiplied by a modulator signal, M, to create a new ring-modulated signal, R:
There are different ways to implement this; most likely it is suitable to simply multiply the two signals, but alternatively the amplitude input of a carrier module can be the output of a modulator module, as well. The frequency of the modulator signal also plays an important role in the characteristic of the RM signal. From this, we achieve the following important result:
If the frequency of M is under 20 Hz or so, we will generally perceive the tremolo effect, where the amplitude of C will vary at the frequency of M. Periodic signals M with a frequency below 20 Hz are called low-frequency oscillators.
When the frequency of M is in the audible range, that is, 20 Hz or more, there is an effect on the timbre of the signal. The variations in amplitude become fast enough that the modulator generates a set of frequency sidebands. With two sine waves as carrier and modulator, RM will generate a frequency spectrum containing two sidebands, which are the sum and difference of the carrier and modulator frequencies. When this occurs, the actual carrier frequency is removed from the spectrum, leaving two harmonic sidebands (if the frequency of C and M are an integer ratio to one another) or two inharmonic sidebands (if the ratio is otherwise). For instance, if the carrier is 900 Hz and the modulator is 500 Hz, we will get two sidebands; one at 400 Hz (900 - 500) and one at 1400 Hz (900 + 500).
If C and M are not sine waves (i.e. their waveforms are more complex) then the resultant signal will contain more than one or many different sidebands at different frequencies and amplitudes, indicating a more complex sound. Figure 8.2 illustrates two examples of ring modulation - the original example with frequencies C = 900 and M = 500 but also when C = 400 and M = 1000, which introduces negative frequencies into the spectrum. This results in a "wrapping" phenomenon where the difference sideband of C and M is -600 Hz! As a result, we find that a difference sideband occurs at 600 Hz, and that is true for any negative frequency; a sideband will occur at its the unsigned (positive) frequency.
Amplitude Modulation 
Amplitude modulation is similar to ring modulation, except it works with a unipolar modulator. The amplitude of the carrier signal, C, is modulated by the unipolar modulator M. At infrasonic (below 20hz) frequencies, the modulator serves to attenuate or boost the amplitude of the signal. A simple example of this would be a typical ADSR envelope, which scales the amplitude of the carrier signal over time from 0 to 1 and back down again. However, for synthesis techniques we generally consider the effect that periodic modulator signals above 20 Hz have on a carrier. Once again, the mathematical form is simply the product of two signals:
Where C is the carrier signal and M is a unipolar modulator, typically set to vary between values of 0 and 1. Without mention of the unipolar modulator, this technique would appear to be identical to ring modulation. Like ring modulation, amplitude modulation produces a pair of sidebands for every sinusoidal component in the carrier and modulator, and these sidebands are generated at frequencies the sum and difference of the two signal frequencies. The difference between the two techniques is highlighted here:
The difference between amplitude modulation and ring modulation is that in AM the carrier frequency is preserved and the sidebands generated are at half the amplitude of the carrier amplitude.
One of the advantages of AM, like its cousin, RM, is that using just two signals or oscillators, we can create some partially rich signals. Using a harmonically dense signal such as a square wave oscillator can create a wealth of sidebands from a minimum of control parameters and computation. Control over these generated partials may not, however, be as detailed and straightforward as techniques such as additive synthesis. As a result, we find that AM and RM is used more often in signal processing than signal generation.
Expanding on amplitude modulation requires us to introduce more parameters and elements to the technique to give it some "weight" with other, more popular techniques. For instance, we can introduce into the system a unipolar low-frequency oscillator which is set to control the amplitude of the modulator; by changing the amplitude of the modulator we are modifying what is known as the modulation index; a factor which controls the strength of the AM sidebands. In addition to modifying the amplitude of the modulator, we can also modify the frequency of the modulator. As you may expect, this causes a shift in the frequencies of the sidebands generated through the AM process and can, carefully controlled, produce some interesting, dynamic sounds that are hard to produce with other techniques. Breaking away from sinusoidal oscillators in both cases of carrier and modulator, and even the modulation of the modulation index is one of the first steps to exploring this technique; try experimenting with the waveshapes introduced in the previous chapters.