Modulation
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Modulation
Modulation is defined as varying one signal according to changes in a second signal. The two signals are referred to as the carrier and the modulator. When dialing into a program on the radio, the frequency we dial into is the carrier, and the radio receiver has to demodulate the signal in order to play it back. While demodulation isn't particularly useful for music, modulation is. Slow amplitude variation creates a tremolo effect, whereas slow frequency variation creates a vibrato effect in traditional instruments and voices. When the frequency of the modulation rises into the audio bandwidth, audible sidebands begin to appear, which are added to the spectrum of the carrier.
The largest benefit of Modulation Synthesis is that it requires much less parameter data, memory, and computation time than additive or subtractive synthesis, but achieves the same complexity of spectrum.
Amplitude Modulation (AM)
AM is the less complicated form of modulation, and unsurprisingly involves modulating the amplitude of the carrier signal. There are three different types of AM:
Type 1: Ring Modulation:
Ring Modulation can also be referred to as balanced, or double-sideband suppressed carrier, but ring is a bit catchier. Ring Modulation is achieved with a simple multiplication of two bipolar audio signals:
f1 ( t ) * f2 ( t )
Ring Modulation suppresses the carrier signal, and creates sidebands of the positive and negative modulator frequency centered around the carrier frequency. If the carrier and modulation signal are in an integer ratio from each other, the sidebands generated are harmonic; otherwise they are inharmonic. If the modulating frequency is higher than the carrier frequency, negative frequencies occur (e.g. for C=100, and M=400, C-M=-300). These negative frequencies create a change in phase of the signal.
Type 2: AM Double Sideband:
In AM Double Sideband Modulation, the carrier is not suppressed, so this type of modulation is useful for radio applications where a receiver has to lock in on the carrier signal in order to tune in a radio station. This type of AM is created by modulating a bipolar carrier signal (ranging from -1 to +1)with a unipolar modulator (ranging from 0 to 1).
.5 * (1. + f1 ( t ) ) * f2 ( t )
carrier: f1 ( t ) = sin ( 2 π fc t) modulator: f1 ( t ) = sin ( 2 π fm t)
Overmodulation occurs when a modulation index of > 1 is used. This starts bleeding over into other frequencies, which is bad for radio, but interesting for music. The rapid on/off cycling of a sound is particularly attractive to the ear.
When an analog amplifier gets a negative value, it just turns off, so overmodulation distorts the program signal and adds harmonics. This doesn't occur automatically with a digital amplifier though, so any value < 0 must be replaced with 0 to emulate an analog amplifier.
Type 3: AM Single Sideband Suppressed Carrier:
Developed by Harald Bode. Also called frequency shifting (or klangumwandler), it has separate outputs for the sum and difference frequencies. Given the limited amount of headroom in a digital signal, this is difficult to replicate digitally.
Frequency Modulation (FM)
Frequency Modulation modulates the frequency of a carrier signal with a modulator signal.
One thing to remember while reading through this section is that Modulation Depth, Frequency Deviation, Modulation Index, MI, and I are all the same...just to keep things interesting I guess.
When the carrier and modulator signal are both sine waves, the formula for a frequency modulated signal at time t is:
FMt = A * sin(Ct + [ I * sin ( Mt ) ] )
A = peak amplitude of the carrier Ct = 2 π * Carrier Mt = 2 π * Modulator I = Modulation Index
f inst = fc + Fdv sin ( 2 π fm t ) a = A sin ( 2 π f inst t )
f inst = instantaneous frequency Fdv = frequency deviation
The number of sidebands generated depends on the amount of modulation applied to the carrier, and the amplitudes of the individual sideband components vary according to Bessel functions of the first kind and nth order, where the argument to the function is the modulation index (MI).
The position of the sidebands generated by FM depends on the ratio of the carrier frequency to the modulating frequency (C:M Ratio). The C:M ratio is also frequently called the Harmonicity Ratio. When C:M is a simple integer ratio, such as 4:1, FM generates harmonic spectra (sidebands are integer multiples of the carrier and modulating frequencies). When C:M is not an integer ratio, such as 8:2.1, FM generates inharmonic spectra. The harmonic spectra is generated at multiples of the sum and difference frequencies (C+M, C+2M, C+3M, C-M, C-2M, C-3M).
The bandwidth of the FM spectrum (the number of sidebands) is controlled by the modulation index (MI), which is defined as I = D / M. D is the amount of frequency deviation from the carrier frequency, and M is the Modulator frequency. The Modulation Index, which is also called the Modulation Depth, is controlled by the amplitude of the modulator signal. When the modulator signal is set to sub-audio frequencies, the increase in its amplitude is heard as an increase in the range of the vibrato caused by the modulator. As a general rule, the number of significant sideband pairs is I+1, and the total bandwidth is equal to twice the sum of D and M.
Some basic properties of FM synthesis ([url]http://x.i-dat.org/~csem/UNESCO/5/[/url]):
Case 1: if ƒc is equal to any integer and ƒm is equal to 1, 2, 3 or 4, then the resulting timbre will have a distinctive pitch, because the offset carrier frequency will always be prominent. Case 2: if ƒc is equal to any integer and ƒm is equal to any integer higher than 4, then the modulation produces harmonic partials but the fundamental may not be prominent. Case 3: if ƒc is equal to any integer and ƒm is equal to 1, then the modulation produces a spectrum composed of harmonic partials; e.g. the ratio 1:1 produces a sawtooth-like wave. Case 4: if ƒc is equal to any integer and ƒm is equal to any even number, then the modulation produces a spectrum with some combination of odd harmonic partials; e.g. the ratio 2:1 produces a square-like wave. Case 5: if ƒc is equal to any integer and ƒm is equal to 3, then every third harmonic partial of the spectrum will be missing; e.g. the ratio 3:1 produces narrow pulse-like waves. Case 6: if ƒc is equal to any integer and ƒm is not equal to an integer, then the modulation produces non-harmonic partials; e.g. 2:1.29 produces a "metallic" bell sound
If modulation goes below 0, we get folding around 0 Hz, and we hear the negative frequencies as if they were positive.
Digital Frequency Modulation:
In the digital domain, we get a linear frequency distribution, so if the carrier and modulator are the same, harmonic sounds are created. This does not occur in the analog domain. With analog FM, we usually get an exponential frequency distribution because of how analog oscillators are designed. A modulating analog oscillator that varies between -1 volt and +1 volt causes a carrier oscillator set to A440 to vary between A220 and A880. This means that it modulates 220 Hz down, and 440 Hz up, which is an asymmetrical modulation. When this occurs, the perceived center pitch is detuned by a significant interval because the average center frequency changes.
Linear frequency modulation allows the creation of sounds that would take large banks of analog oscillators with additive synthesis. This use of digital FM was patented at Stanford by John Chowning, and licensed to Yamaha in the 1970s for the DX7 synthesizer.
A quote from John Chowning:
Then in about 1970 I remembered some work that Jean-Claude Risset had done at Bell Labs, using a computer to analyze and resynthesize trumpet tones. One of the things that he realized in that work is that there is a definite correlation between the growth of intensity during the attack portion of a brass tone and the growth of the bandwidth of the signal. For the first few milliseconds, what energy is there is mostly around the fundamental; and quickly, as the intensity grows during the next 30 or 40 milliseconds, more and more harmonics appear at a successively higher volume. I thought about that, and I realized that I could do something similar with simple FM, just by using the intensity envelope as a modulation index. That was the moment when I realized that the technique was really of some consequence, because with just two oscillators I was able to produce tones that had a richness and quality about them that was attractive to the ear - sounds which by other means were quite complicated to create. For example, Jean-Claude had to use 16 or 17 oscillators to create a similar effect using additive synthesis techniques.
C:M Ratios
A clarinet can be reproduced with a C:M Ratio 3:2. The 3 in the ratio tells us that there will be more energy in the third harmonic at the beginning of the tone (when the FMI is still at 0 in the figure below). The 2 in the ratio tells us that every even harmonic will sound, but no odd harmonics.
Enharmonic Percussion Sounds are characterized by non-integer C:M ratios, such as 1:1.4 for a wood drum. The FMI changes rapidly in the figure below because rapid changes give a "wooden" texture to the sound.
Detuning can be accomplished with very small C:M ratios, such as 1000:998.
.5 Hz difference => 1 Hz beat 1 Hz difference => 2 Hz beat 2 Hz difference => 4 Hz beat
Multiple Carrier FM
Like the name sounds, Multiple Carrier Frequency Modulation uses more than one carrier, but only one modulator signal. Multiple carriers create formant regions (peaks) in the spectrum, as shown below. The presence of formant regions is characteristic of the human voice, and most traditional instruments. It is also possible to set separate decay times for each formant, which can be useful for creating brass sounds where the upper partials decay more rapidly than lower partials.
Most applications of Multiple Carrier FM try to simulate the sounds of traditional instrument or vocal tones. Chowning applied the technique to the synthesis of vowel sounds sung by a soprano and by a bass voice. He determined that a combination of periodic and random vibrato must be applied to all frequency parameters for realistic simulation of vocal tones. Chowning's vibrato percent deviation V is defined as:
V = 0.2 * log(pitch)
For example, a pitch of 440 Hz would have a vibrato of about 1.2 percent, which equals 5.3 Hz in depth.
Multiple Modulator FM
[center](aka FM Complex Modulating Wave)[/center]
In Multiple Modulator Frequency Modulation more than one oscillator modulates a single carrier oscillator. The two basic configurations of this are parallel and series.
In Parallel MM FM, two modulator signals simultaneously modulate a single carrier sine wave. The modulation generates sidebands at frequencies of the form:
C ± (i * M1) ± (k * M2)
i, k are integers M1, M2 are modulating frequencies
Series MM FM modulates the modulator with another modulator before modulating the carrier. This variety of FM helps to get beyond the characteristic bessel function sound of FM. It's typically impossible to get a flat spectrum with FM without modulating the modulator.
Sidebands are produced at:
fc ± i fm1 ± K fm2
One fm pair is modulated by another
(fc ± i fm1) ± K fm2
Feedback Frequency Modulation
Feedback FM is widely used, and its application in synthesizers is patented by Yamaha. It solves certain problems associated with simple FM methods. The undulation of the amplitude of partials according to Bessel Functions creates an unnatural "electronic sound", and feedback FM makes the spectrum more linear in its evolution: as the modulation index increases, the number of partials and their amplitude increases relatively linearly.
The first feedback oscillator instrument was used by Jean-Claude Risset in 1969, but the technique first appeared in public in the paper, "Some idiosyncratic aspects of computer synthesized sound" by Arthur Layzer. In this paper, Layzer described work in developing a self-modulating oscillator whose output is fed back into its input. The main difference between early feedback oscillators and the Yamaha feedback oscillator is that the early feedback oscillators were implementing a form of feedback AM by feeding the output back into the amplitude input, whereas Yamaha fed the output back into the frequency input, which is feedback FM.
One-Oscillator Feedback:
One-Oscillator Feedback FM works by feeding its output back into its frequency input through a multiplier and an adder. The multiplier controls the feedback factor, which works as a scaling function for the feedback. As the feedback factor increases, the number of partials and the differences in amplitudes between the partials contribute to a quasi-linear spectral buildup. The adder computes the phase index for the sine table lookup within the oscillator. In a synthesizer, this value is usually obtained by pressing a key on the keyboard which will send a large phase increment value for a high-pitched note, or a small value for a low-pitched note.
Two-Oscillator Feedback:
In Two-Oscillator, the output of a feedback FM Oscillator (using one-oscillator feedback fm) is used to modulate a second, nonfeedback oscillator. The multiplier at the inlet of the nonfeedback oscillator (M) acts as an index of modulation control between the two oscillators. When M is between 0.5 and 2, the amplitude of the partials decreases as the number of partials increases.
Three-Oscillator Indirect Feedback:
Indirect feedback produces a complex form of modulation. Basically, it is designed as a chain of three oscillators, with the output of the carrier oscillator fed back into the first modulation oscillator. When the frequencies of the three oscillators are noninteger multiples, nonpitched sounds are created. A beating chorus effect is produced when these frequencies are very close to being an integer relationship.
Frequency Modulation Conclusions
In the end, it turns out that while FM can create wonderfully complex tones that can frequently fool the ear into categorizing a sound into particular acoustic instrument categories, it doesn't really create sounds that sound exactly like any particular instrument. It does have some potentially strong musical uses, but it can't be used to generate any possible sound, and can't be used for everything.
Discrete Summation Formulas (DSFs)
James A. Moorer showed that the equation for single FM is one instance of a general class of equations called Discrete Summation Formulas. DSFs are a set of formulas that are closed form, which means that they are a compact and efficient representation of a longer summation formula. With DSFs, we have only a few parameters to manipulate, which makes it possible to realize in a digital form. While most are useless musically, some other DSFs can generate time-varying tones similar to FM, some can generate one-sided spectra whose partials extend just in one direction from the carrier frequency, and some can generate tones whose partial amplitudes increase instead of decrease by a constant factor.
