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FM Synthesis
Anders Ćland
Roger B. Dannenberg
Introduction to Computer Music
Carnegie Mellon University
1 Introduction
Frequency modulation (FM) is a synthesis technique based on the simple idea of periodic modulation of the signal frequency. That is, the frequency of a carrier sinusoid is modulated by a modulator sinusoid. The peak frequency deviation, AKA depth of modulation, expresses the strength of the modulatorās effect on the carrier oscillatorās frequency. FM synthesis was invented by John Chowning (1973), and became very popular due to its ease of implementation and com- putationally low cost, as well as its (somewhat surprisingly) powerful ability to create realistic and interesting sounds. To begin with, letās look at the equation for a simple frequency controlled sine oscillator. Often, this is written
y(t) = A sin(2Ļf t) (1)
where f is the frequency in Hz. However, this only works for fixed frequency and amplitude. To deal with a time-varying frequency, we must integrate the frequency function f to determine the accumulated phase at time t:
y(t) = A(t) sin(
ā« (^) t
0
2 Ļf (x)dx) (2)
Frequency modulation uses a rapidly changing function
f (t) = C + D sin(2ĻM t) (3)
where C is the carrier, a frequency offset that is in many cases is the fundamental or āpitchā. D is the depth of modulation that controls the amount of frequency deviation (called modulation), and M is the frequency of modulation in Hz. Plugging this into equation 2 and simplifying gives the equation for FM:
f (t) = A sin(2ĻCt + D sin(2ĻM t)) (4)
Note that this equation is not exactly right. We assume that the phase of the modulation does not matter. Thus, while the integral of the sin function is cos, we keep the sin term because sin is the same as cos with a phase shift. In practice, the phase can make a subtle difference, but nearly everyone ignores it. I = DM is known as the index of modulation. When D 6 = 0, sidebands appear in the spectra of the signal; above and below the carrier frequency C, at multiples of Ā±M. In other words, we can write the set of frequency components as C Ā± kM , where k=0,1,2,.... The number of significant components increases with I, the index of modulation.
2 Negative Frequencies
According to these formulas, some frequencies will be negative. This can be interpreted as merely a phase change: sin(āx) = ā sin(x) or perhaps not even a phase change: cos(āx) = cos(x). Since we tend to ignore phase, we can just ignore the sign of the frequency and consider negative frequencies to be positive. We sometimes say the negative frequencies āwrap aroundā (zero) to become positive. The main caveat here is that when frequencies wrap around and add to positive frequencies of the same magnitude, the components may not add in phase. The complexity of all this tends to give FM signals a complex behavior as the index of modulation increases, adding more and more components, both positive and negative.
3 Harmonic Ratio
The human ear is very sensitive to harmonic vs. inharmonic spectra. Perceptu- ally, harmonic spectra are very distinctive because they give a strong sense of pitch. The harmonic ratio [Truax 1977] is the ratio of the modulating frequency to the carrier frequency, such that H = MC. If H is a rational number, the spectrum is harmonic; if it is irrational, the spectrum is inharmonic.
3.1 Rational Harmonicity
If H = 1 the spectrum is harmonic and the carrier frequency is also the fun- damental, i.e. F 0 = C. To show this, remember that the frequencies will be C Ā± kM , where k=0,1,2,..., but if H = 1, then M = C, so the frequencies are C Ā± kC, or simply kC. This is the definition of a harmonic series: multiples of some fundamental frequency C. When H = (^) m^1 , and m is a positive integer, C instead becomes the māth component (harmonic) because the spacing between harmonics is M = C/m, which is also the fundamental: F 0 = M = C/m. With H = 2, we will get sidebands at CĀ± 2 kC (where k=0,1,2,...), thus omitting all even harmonics - which is ideal for modeling a clarinet.
5 Index of Modulation
The index of modulation, I = (^) MD , allows us to relate the depth of modulation, D, the modulation frequency, M , and the index of the Bessel functions. In practice, this means that if we want a spectrum that has the energy of the Bessel functions at some index I, with frequency components separated by M , then we must choose the depth of modulation according to the relation I = (^) MD [F. R. Moore 1990]. As a rule-of-thumb, the number of sidebands is roughly equivalent to I + 1. That is, if I = 10 we get 10 + 1 = 11 sidebands above, and 11 sidebands below the carrier frequency. In theory, there are infinitely many sidebands at C Ā± kM , where k=0,1,2,... if the modulation is non-zero, but the intensity of sidebands falls rapidly toward zero as k increases, so this rule of thumb considers significant sidebands.
6 Nyquist & FM
In Nyquist we can use the built-in function
fmosc(pitch, modulation, table, phase)
for FM synthesis - about which the manual says: āReturns a sound which is table oscillated at pitch plus modulation for the duration of the sound modulation.ā The table and phase parameters are optional and often omitted: the default table is a sinusoid, and the initial phase generally does not change the resulting sound. When you create an FM instrument, keep in mind exactly how the modula- tion parameter given to fmosc() relates to the FM equation, f (t) = A sin(2ĻCt+ D sin(2ĻM t)). Namely, that modulation denotes the term, modulation = D sin(2ĻM t).
6.1 Example
Produce a harmonic sound with about 10 harmonics and a fundamental of 100 Hz. We can choose C = M = 100. Since the number of harmonics is 10 we need 9 sidebands, and so I +1 = 9 or I = 8. I = (^) MD or D = IM , so D = 8ā100 = 800. Finally, we can write fmosc(hz-to-step(100), 800 * hzosc(100)).
7 Examples of FM Signals
Figures 2 & 3 show examples of FM signals. The X-axes on the plots represent time - here denoted in multiples of Ļ.
Figure 2: A = 1, C = 242 Hz, D = 2, M = 40 Hz.
Figure 3: A = 2, C = 210 Hz, D = 10, M = 35 Hz.
