Computer Music (MUSC 216)
FM (Frequency Modulation) Synthesis

See: frequency modulation. Also see fm synthesis and fm synthesis

Frequency Modulation is a technique not just limited to sound synthesis but has been used in radio transmission since around 1935 (Edwin Armstrong, "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation"). See Frequency Modulation. John Chowning discovered the possibilities for sound synthesis in the 1960's while experimenting with modulation techniques at Stanford University. Ordinarily, if an LFO (low frequency oscillator) is set to modulate the frequency of another oscillator, the result is VIBRATO. However, if the frequency of the modulator is increased to above about 20hz (the threshold of human hearing), additional frequencies are perceived and thus the possibilities of frequency modulation for creating complex sounds.

Modulation means using one signal to affect another signal. A simple example of Frequency Modulation synthesis (FM) consists of one oscillator (the MODULATOR), modulating the frequency of another oscillator (the CARRIER). This procedure produces additional frequencies called SIDEBANDS which form symmetrically around the original frequency of the CARRIER. For example, here is a simple SYD FM patch:

(Figure 1: Simple FM patch using SYD)

When synthesized, this patch will produce the following waveform:

(Figure 2: Output of Simple FM patch from Fig. 1)

A spectral analysis of the wave shows the sideband frequencies beginning to emerge symmetrically around the original Carrier frequency. Although clearly visible in the spectrum, the sideband frequencies have only slightly altered the original sine wave.

(Figure 3: Spectrum of Simple FM patch from Fig. 1)

In this particular carrier/modulator ratio (c= carrier; m = modulator), the sideband frequencies form at c+ m and c-m. Additional sets of frequencies should emerge as the modulation amount increases.

Increasing the modulation amount should “steal” more energy away from the carrier (thus reducing its amplitude) while increasing the amplitudes of the sideband frequencies. Also addition sideband frequencies should appear at c + 2m and c -2m). Here is the same SYD patch with a modulation amount at 300 Hz.

(Figure 4: Output of Simple FM patch from Fig. 1 with a Modulation Amt. of 300 Hz.)

There is a much more noticeable alteration of the wave form. The spectrum will show more sideband frequencies emerging. In addition, the amplitudes of the sideband frequencies increase as they steal energy away from the carrier frequency.

Here is a spectral analysis:

(Figure 5: Spectrum of Simple FM patch from Fig. 1 with a Modulation Amt. of 300 Hz)

At a modulation amount of 500 Hz. there is a significant alteration of the waveform:

(Figure 6: Output of Simple FM patch from Fig. 1 with a Modulation Amt. of 500 Hz.)

The spectrum reveals that most of the amplitude energy as been stolen from the carrier frequency as more sideband frequencies emerge.

(Figure 7: Spectrum of Simple FM patch from Fig. 1 with a Modulation Amt. of 500 Hz)

If the original CARRIER contains additional partials (it is a more complex sound than just a simple waveform) then SIDEBANDS form in pairs around each partial. The intensity of the SIDEBAND frequencies is governed by the MODULATION INDEX which is in essence the depth or amount of the modulation. The actual MODULATION INDEX (I) is the ratio of the frequency of the modulating signal (mf) to the amplitude of the modulating signal (ma) -- as in ma/mf. The basic FM process may be described by the formula:

e(t) = A sin[ct + I sin(mt)], where:

e(t) is the FM output

A is the Amplitude

c is the carrier wave frequency

m is the modulation frequency

I is the modulation amount (depth of modulation)

Sidebands are produced at frequencies c, c + m, |cm|, c+2m, |c-2m|, c + 3m, |c - 3m|, etc. The actual number of sideband frequencies is related to the MODULATION AMOUNT (I) such that as I increases, energy is "stolen" from the carrier and distributed among an increasing number of sideband frequencies, thus increasing the BANDWIDTH. Another way of writing this formula is:

| C n m | n = 0,1,2,3, . . . . . where n is the sideband pair number. In this expression, the + indicates the upper sidebands and the - indicates the lower sidebands.

The amplitudes of the carrier and sideband components are determined by Bessel functions of the first kind and the nth order, Jn(I), the argument to which is the MODULATION AMOUNT (I). The zeroth-order Bessel function and amount J0(I), yields a scaling coefficient for the first upper- and lower-side frequencies; the second-order J2(I), for the second upper- and lower-side frequencies; and so forth. The higher the order of the side frequency the larger the modulation amount must be for that side frequency to have any significant amplitude. The total bandwidth is approximately equal to twice the sum of the FREQUENCY DEVIATION and the modulation frequency, or

BW ~ 2(d + m) where d is the FREQUENCY DEVIATION or amplitude of the modulating signal.

The FREQUENCY DEVIATION is equivalent to the AMPLITUDE of the modulating signal. Consequently, the stronger the AMPLITUDE of the modulating signal, the greater the BANDWIDTH (the greater number of sidebands) in the resulting SPECTRUM. [See Computer Music, 2nd ed, Charles Dodge, Schirmer Books, 1997, p. 116). "The distribution of power among the spectral components depends ... on the amount of frequency deviation ... produced by the modulating oscillator.... Increasing the deviation causes the sidebands to acquire more power at the expense of the power in the carrier frequency. The wider the deviation, the more widely distributed is the power among the sidebands and the greater the number of sidebands that have significant amplitudes. Thus the deviation can act as a control on the bandwidth of the spectrum of an FM signal.

The amplitude of each spectral component is determined by BOTH THE 'DEVIATION' (modulation amount) AND THE FREQUENCY OF MODULATION. To describe these amplitudes mathematically, it is useful to devine an INDEX OF MODULATION:

I = d/fm [ Index(I) = the relationship of the frequency deviation (d) -- or "modulation amount" -- to the frequency of modulation (fm) ]

... or for our purposes, the relationship of the modulation amount (or 'deviation') (ma) to the modulation frequency (mf)

I = ma/mf

See Charles Dodge, p. 117.

The amplitude (and position) of each sideband frequency depends on the index of modulation where the actual frequency of a particular sideband is symmetrically related to the carrier frequency. For example, if the carrier frequency is 440 and the modulation frequency is 100 then the sidebands will be symmetrically arranged both above and below the carrier frequency accordingly at both +100 and -100 increments: 440 +100; 440 - 100; 440 +200; 440 -200; 440 +300; 440 - 300, etc. The STRENGTH or amplitude of the individual sidebands will be dictated by the modulation index. So when the value of the modulation amount increases, it gives power (amplitude) to the sideband freuqencies by "stealing" power (amplitude) away from the carrier frequency. Consequently as the sideband frequencies increase in amplitude, the carrier frequency decreases in amplitude.

The MODULATION AMOUNT is a value that is added to the Carrier FREQUENCY over time in both a positive and negative value at a RATE that is dictated by the modulation frequency. If the MODULATION AMOUNT is a large value such that it is GREATER than the Carrier Frequency, then NEGATIVE FREQUENCIES may be generated, creating in effect -- digital NOISE, unless the hardware or software compensates for the negative frequency values. For example, if the Carrier Frequency is 300 Hz and the Modulation Amount is 200HZ, then the lowest frequency that will result in the modulatin process = 300 - 200 (no problem!). However if the Carrier Frequency is 300 Hz and the Modulation amount is 500 Hz, then the lowest frequency that will result in the modulation process = 300 - 500 (big problem unless the hardware or software compensates for the negative frequencies, for example by utilizing the ABSOLUTE VALUE of the modulation amount). JSYD DOES compensate for the negative values by automatically using the absolute value of the modulation amount.

As the sideband frequencies grow in amplitude and consequently distribute themselves symmetrically around the carrier frequency, eventually the sideband frequencies to the LEFT of the carrier frequency will reach a frequency value less than ZERO. If sideband frequencies appear as negative values, then they are in REALITY positive frequencies with a PHASE SHIFT of 180 degrees which results in 'fold-over', especially where these low frequencies coincide. The practical result is that as the modulation amount is increased more and more, the low frequency amplitudes increase more and more often giving unexpected results. See FM Index.

Ordinarily, these negative signs are ignored in plotting spectra. However, in the application of FM, this phase information is SIGNIFICANT and must be considered in plotting spectra. Because of the change in phase, if a shifted sideband frequency coincides with an existing sideband frequency, the resulting amplitude will be the sum of their amplitudes taking into account the effects of constructive interference.

See Computer Music: Synthesis, Composition, and Performance by Charles Dodge and Thomas A. Jerse (New York: Schirmer Books, 1985, 1997, pp. 115 - 139).

Also see this excellent explanation from Indiana University's Electronic Music site: Principles of Frequency Modulation

"The maximum instantaneous frequency that the carrier oscillator will assume is [c + m] and the minimum is [c - m]. If the deviation is large due to an increase , it is possible for the carrier oscillator to have a negative number applied to its frequency input (see above under MODULATION INDEX ). In a digital oscillator, this corresponds to a negative sampling increment, forcing the oscillator's phase to move backward. Most, but not all, digital oscillators are capable of doing this (fold-over does not happen). Those that cannot are of limited usefulness for FM synthesis because the maximum deviation cannot exceed the carrier frequency and consequently result in large amounts of unusable digital noise." [Dodge, pp. 116]

When creating an FM patch with JSYD, this problem is avoided because JSYD automatically takes the ABSOLUTE VALUE of the modulation frequency and adds it to the carrier frequency.

Check out these links:

Synthesis Facts

Digital Audio Facts

Additive Synthesis

Midi Note Number to Equal Temperament Semitone to Herz Conversion Table

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