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Introduction to FM (Frequency Modulation) Synthesis
Written by John Duesenberry - © 2005, PRIMEDIA Business Magazines & Media Inc. Reprinted with Permission.
FM synthesis, in a rather limited form, is possible with voltage-controlled analog synthesizers. But the musical potential of FM didn’t become fully apparent until John Chowning’s pioneering work on the digital implementation of FM in the 1970s. A decade later, when Yamaha introduced the DX7 synthesizer and its many relatives, FM won mass acceptance in the music world.

The FM craze of the 1980s has abated, but one industry pundit, impressed by Yamaha’s new FS1R synth, recently predicted an FM synth revival. If he’s right, then we’ve picked a good time to reexamine the subject.

Boot Camp Revisited
For those who missed (or forgot) the earlier article, let’s start with a rapid review of the general characteristics of modulation synthesis. Modulation synthesis is a waveshaping technique in which an audio-rate signal called the modulator controls some parameter of another audio signal, called the carrier. In FM, the frequency is the modulated parameter. The modulation process generates new sine-wave components, called sidebands, in the spectrum of the output signal. The power of the sidebands is governed by the modulation index (discussed shortly). The index is defined differently for AM and FM, but in both cases it is related to the amplitude of the modulator.

Sideband frequencies can be calculated by taking the sums and differences of the frequencies of carrier and modulator components. The resultant spectra fall into two broad classes. In a harmonic spectrum, all components are members of a harmonic series, that is, they are integer multiples of some fundamental frequency. In other words, their frequencies are 2, 3, 4, and so on, times the frequency of the fundamental. In an inharmonic spectrum, some or all components do not fit into a harmonic series.

The ratio of the carrier and modulator signals’ frequencies, which can be represented as Fc:Fm, determines whether a modulation spectrum will be harmonic or inharmonic. Here we repeat Rule 1 for fledgling modulation synthesists:

Rule 1. If Fc:Fm is a ratio of simple integers, the modulation spectrum will be harmonic. Otherwise, the spectrum will be inharmonic.

Now let’s relate these generalities to the specifics of FM.

Simple FM Spectra
One reason for FM’s popularity is that interesting spectra can be synthesized with limited resources. In FM, a sine carrier and modulator generate a theoretically infinite number of sidebands. By varying one simple parameter, the modulation index, we can create complex variations in the spectrum. Sine-wave FM with dynamic index control (that is, an index that changes over time) is the basis of most commercial FM synthesizers.

Figure 1 shows the output waveform that is created from a typical sine carrier and modulator. The waveshaping effect, a kind of “bending” of the sine waveform as its instantaneous frequency changes, is apparent. Though this may be visually interesting, the spectrum that results is even more so.

The resulting spectrum of an FM process is easy to predict. Given a sine carrier of frequency Fc and a sine modulator of frequency Fm, the FM spectrum will consist of the following components:

Upper sideband frequencies, which are the sum of Fc and every integer multiple of Fm (Fc + Fm, Fc + 2Fm, Fc + 3Fm, and so on).

Lower sideband frequencies, which are the difference of Fc and every integer multiple of Fm (Fc – Fm, Fc – 2Fm, Fc – 3Fm, and so on).

The original carrier frequency, Fc.

Let’s consider the spectrum resulting from FM with a 500 Hz carrier and a 400 Hz modulator. The ratio Fc:Fm reduces to 5:4, so according to Rule 1, this will be a harmonic spectrum. Figure 2a represents the first three sideband pairs around the carrier. Notice that some lower sidebands have negative frequencies. This will occur whenever Fm or one of its multiples is greater than Fc. A signal with frequency –F is simply inverted (180 degrees out of phase) with respect to a frequency F. In this particular example, the negative frequencies wouldn’t affect the sound.

In Figure 2b, negative components are represented as having negative amplitudes (downward lines). In fact, they are positive, but 180 degrees out of phase, as noted earlier. This makes it easy to see that the spectrum is harmonic, as predicted. It consists of a 100 Hz fundamental, with odd-numbered harmonics. Don’t confuse the carrier component (500 Hz) with the fundamental (a 100 Hz sideband).

Figure 2c shows how negative frequencies affect the sound. This is another harmonic spectrum, because Fc and Fm are both 100 Hz (Fc:Fm = 1:1). Although the sideband series around the carrier extends infinitely, only the first three sideband pairs are shown. The first lower sideband has a frequency of 0, which is an inaudible DC component. As the brackets show, the 100 Hz carrier component is matched with a sideband of its inverted frequency (–100 Hz), as is the 200 Hz component.

The brackets indicate a pattern that holds throughout this spectrum: for each positive, nonzero component, there is a corresponding negative component of unequal amplitude. The summation of the corresponding positive/negative components produces a partial cancellation, or attenuation, of every positive component in the spectrum. Figure 2d illustrates this result. The audible spectrum has a fundamental of 100 Hz, with all harmonics present.

Here’s a quick quiz: compute the first three FM sideband pairs where Fc = 500 Hz and Fm = 202.61. Is this spectrum harmonic or inharmonic? (Answer: The lower sidebands will appear at 297.39, 94.78, and -107.83. The upper sidebands are 702.61, 905.22, and 1107.83. The spectrum is inharmonic.)This is an excerpt from the following article: FM Basic Training.

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