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We once again fall back on trigonometry: , but , so the negative frequency component adds to the positive frequency component if it's a cosine, but subtracts if it's a sine.

We get a different pattern of cancellations depending on the initial phases of the carrier and modulator. We're using phase-modulation for simplicity the integration in FM changes the effective initial phase. By varying the relative phases, we can get a changing spectrum from these cancellations. Unfortunately, in this case there are some pesky -1's floating around, so we get asymmetric or gapped spectra, but not anything we'd claim was single side-band.

It produces beautiful single-sided spectra. We might grumble that the sideband amplitudes don't leave us much room for maneuver, but the factorial in the denominator overwhelms any exponential in the numerator, so we can get many interesting effects: moving formants, for example. Here we sweep "r" from Here again trigonometry can be used to expand The modulating signal is now made up of two sinusoids don't despair; this is a terminating sequence.

Since sine is not linear it is , this is not the same thing as In the second case we just add together the two simple FM spectra, but in the first case we get a more complex mixture involving all the sums and differences of the modulating frequencies. These sum and difference tones "intermodulation products" are not limited to FM.

Any nonlinear synthesis technique produces them. The result can be expressed: You can chew up any amount of free time calculating the resulting side band amplitudes — see the immortal classic: Schottstaedt, "The Simulation of Natural Instrument Tones Using Frequency Modulation with a Complex Modulating Wave".

There's a function to do it for you in dsp. In simple cases, the extra modulating components flatten and spread out the spectrum somewhat see below and ncos for discussions of very different not-so-simple cases. By using a few sines in the modulator, you get away from the simple FM index sweep that has become tiresome, and the broader, flatter spectrum is somewhat closer to that of a real violin.

Since we can fiddle with the initial phases of the modulating signal's components, we can get very different spectra from modulating signals with the same magnitude spectrum. In the next two graphs, both cases involve a modulating signal made up of 6 equal amplitude harmonically related sinusoids, but the first uses all cosines, and the second uses a set of initial phases that minimizes the modulating signal's peak amplitude: cascade FM: sin sin sin We can, of course, use FM or anything to produce the modulating signal.

Our fundamental frequency no longer has any obvious relation to! For example, if we have oscil gen 0. If you are using low indices and the top pair's mc-ratios are below 1. The middle FM spectrum will then have only sines not cosines , so the DC component will be thoroughly discouraged. Or use phase modulation instead; in that case, we have effectively oscil gen 0. Here's an experiment that calls sin sin sin The result is a much broader, flatter spectrum than you normally get from FM.

If you just push the index up in normal FM, the energy is pushed outward in a lumpy sort of fashion, not evenly spread across the spectrum.

In effect we've turned the axis of the Bessel functions so that the higher order functions start at nearly the same time as the lower order functions. The new function Jn nB decreases very! Tomisawa suggests that B should be between 0 and 1. Since we are dividing by B in the equation, we might worry that as B heads toward 0, all hell breaks loose, but luckily and for all the other components so, just as in normal FM, if the index is 0, we get a pure sine wave.

Increasing the sampling rate, or decreasing the carrier frequency reduces this component without affecting the others, but low-pass filtering the output does not affect it so it's unlikely to be an artifact of aliasing which is a real problem in feedback FM.

Fiddle with the initial phase in that line, and there's always some choice that reduces it to 0. Tomisawa's picture of the noise Why does an index over 1. Each burst happens as the modulator phase goes through an odd multiple of pi where sine is going negative as the phase increases.

This is confusing to analyze because at this point in the curve, the feedback is already holding the phase back, so we need to reach a point where the increase in the backup overwhelms the increment on that sample, thereby backing up the overall phase beyond its previous held back value.

So the modulator phase backs into the less negative part of the sine curve: our next y value is less negative it can even be positive! We've started to zig-zag down the sine curve. Depending on the index, this bouncing can reach any amplitude, and start anywhere after the high point of the curve. Eventually, the sine slope lessens as it reaches its bottom , the overall phase catches up, and the bouncing stops for that cycle. The noise is not chaos in the sense of period doubling , or an error in the computation.

If we change the code to make sure the carrier phase doesn't back up, the bursts go away until the index reaches about 1. The take-home message is: "keep the index below 1. FM and noise: sin sin rand One way to make noise deliberately with FM is to increase the index until massive aliasing is taking place.

A more controllable approach is to use a random number generator as our modulator. In this case, the power spectral density of the output has the same form as the value distribution function amplitude distribution as opposed to frequency of the modulating noise, centered around the carrier. The bandwidth of the result is about 4 times the peak deviation the random number frequency times its index — is this just Mr Carson again? Heinrich Taube had the inspired idea of feeding the noise as a sort of cascade FM into the parallel modulators of an fm-flute, but not into the carrier.

The modulating signal becomes a sum of two or three narrow band noises narrow because normally the amplitude of the noise is low , and these modulate the carrier. Previously, we could fix up each modulating sinusoid both in amplitude and initial phase , but here we have no such handles on the components of the incoming signal.

FM square-wave is:. I call this "contrast-enhancement" in the CLM package. This is probably best understood by thinking of FM as a spectral modeling technique, as will be illustrated further below. Figure G. FM bandwidth expands as the modulation -amplitude is increased in G. The th harmonic amplitude is proportional to the th-order Bessel function of the first kind , evaluated at the FM modulation index.

Thus, increasing the FM index brightens the tone. Figure: Bessel functions of the first kind for a range of orders harmonic numbers and argument FM index from [ ].

FM Brass Jean-Claude Risset observed , based on spectrum analysis of brass tones [ ], that the bandwidth of a brass instrument tone was roughly proportional to its overall amplitude. In other words, the spectrum brightened with amplitude.

This observation inspired John Chowning's FM brass synthesis technique starting in [ 40 ]. A simple example of an FM brass instrument is shown in Fig. Note how the FM index is proportional to the amplitude envelope carrier amplitude. FM Voice FM voice synthesis [ 39 ] can be viewed as compressed modeling of spectral formants.

Or use phase modulation instead; in that case, we have effectively oscil gen 0. In simple cases, the extra modulating components flatten and spread out the spectrum somewhat see below and ncos for discussions of very different not-so-simple cases. A slightly bizarre sidelight: there's no law against a modulating signal made up of complex numbers.- Business plan aufbau vorlage bewerbung;
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Doesn't the FM version sound richer and, far more importantly, louder? Each burst happens as the modulator phase goes through an odd multiple of pi where sine is going negative as the phase increases. In the next two graphs, both cases involve a modulating signal made up of 6 equal amplitude harmonically related sinusoids, but the first uses all cosines, and the second uses a set of initial phases that minimizes the modulating signal's peak amplitude: cascade FM: sin sin sin We can, of course, use FM or anything to produce the modulating signal. These functions were studied by Daniel Bernoulli the vibrations of a heavy chain, , Euler the vibrations of a membrane, , Lagrange planetary motion, , and Fourier the motion of heat in a cylinder, ; Bessel studied them in the context of Kepler's equation, and wrote a monograph about them in If we use an mc-ratio of.

A simple example of an FM brass instrument is shown in Fig. Any nonlinear synthesis technique produces them. Mr Carson's opinion of FM: "this method of modulation inherently distorts without any compensating advantages whatsoever".

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IRE, vol 10, pp , Feb Or perhaps most forthright, start with the formula for Jn B given above the integral , and say "we want cos sin expanded as a sum of cosines, and we define Jn to be the nth coefficient in that sum". This was the approach of Bessel and other 19th century mathematicians, but it is not very satisfying for some reason.

What if for example , the modulator has the same frequency as the carrier, and its index B is high enough that some significant energy appears at? Our result is now This is Chowning's version of the expansion. Thus, increasing the FM index brightens the tone. In this case, cos is no longer bounded, so the output can peak at anything, but we still get FM-like spectra.

**Aralkree**

So the modulator phase backs into the less negative part of the sine curve: our next y value is less negative it can even be positive! And since the number of significant components in the spectrum is nearly proportional to the index Carson's rule , we can usually predict more or less what index we want for a given spectral result.

**Malkree**

Since sine is not linear it is , this is not the same thing as In the second case we just add together the two simple FM spectra, but in the first case we get a more complex mixture involving all the sums and differences of the modulating frequencies. One of the raspier versions of the fm-violin used a sawtooth wave as the carrier, and some sci-fi sound effects use triangle waves as both carrier and modulator. Here we sweep "r" from To get a changing spectrum, we need only put an envelope on the fm-index: with-sound fm 0 0. The original pioneer patent expired in , but additional patents were filed later. For further reading about FM synthesis , see the original Chowning paper [ 38 ], paper anthologies including FM [ , ], or just about any computer-music text [ , , 60 , ].

**Zukree**

A slightly bizarre sidelight: there's no law against a modulating signal made up of complex numbers. Tomisawa suggests that B should be between 0 and 1. We've started to zig-zag down the sine curve.

**Gukus**

If you just push the index up in normal FM, the energy is pushed outward in a lumpy sort of fashion, not evenly spread across the spectrum. Thus, a sung vowel can be synthesized using only six sinusoidal oscillators using FM. Here's an experiment that calls sin sin sin If we add cosines at the amplitudes given by the Bessel functions using additive synthesis to produce the same magnitude spectrum as FM produces , we get a very different waveform. The Carson quote is also from that paper originally published in Proc. Then they have a bump and tail off as a sort of damped sinusoid: C G J Jacobi, Gesammelte Werke, VI As the index sweeps upward, energy is swept gradually outward into higher order side bands; this is the originally exciting, now extremely annoying "FM sweep".

**Tygogul**

Since sine is not linear it is , this is not the same thing as In the second case we just add together the two simple FM spectra, but in the first case we get a more complex mixture involving all the sums and differences of the modulating frequencies. FM Voice FM voice synthesis [ 39 ] can be viewed as compressed modeling of spectral formants. By using a few sines in the modulator, you get away from the simple FM index sweep that has become tiresome, and the broader, flatter spectrum is somewhat closer to that of a real violin. If you just push the index up in normal FM, the energy is pushed outward in a lumpy sort of fashion, not evenly spread across the spectrum.

**Sabar**

In this case, the power spectral density of the output has the same form as the value distribution function amplitude distribution as opposed to frequency of the modulating noise, centered around the carrier. Tomisawa suggests that B should be between 0 and 1. Next Section:. The modulation index is set to give the desired bandwidth for the formant group.

**Tekazahn**

A slightly bizarre sidelight: there's no law against a modulating signal made up of complex numbers. Depending on the index, this bouncing can reach any amplitude, and start anywhere after the high point of the curve. If you just push the index up in normal FM, the energy is pushed outward in a lumpy sort of fashion, not evenly spread across the spectrum. If we add cosines at the amplitudes given by the Bessel functions using additive synthesis to produce the same magnitude spectrum as FM produces , we get a very different waveform. This is probably best understood by thinking of FM as a spectral modeling technique, as will be illustrated further below. Unfortunately, in this case there are some pesky -1's floating around, so we get asymmetric or gapped spectra, but not anything we'd claim was single side-band.

**Meztishura**

It makes a sound crisper; "Wait for Me! Since we can fiddle with the initial phases of the modulating signal's components, we can get very different spectra from modulating signals with the same magnitude spectrum. FM waveform index: 3. The Carson quote is also from that paper originally published in Proc.

**Fesho**

If the math side of my article is of any interest, you might like Benson's discussion of FM. Increasing the sampling rate, or decreasing the carrier frequency reduces this component without affecting the others, but low-pass filtering the output does not affect it so it's unlikely to be an artifact of aliasing which is a real problem in feedback FM. We have to return to our initial set of formulas. In effect we've turned the axis of the Bessel functions so that the higher order functions start at nearly the same time as the lower order functions. We're using phase-modulation for simplicity the integration in FM changes the effective initial phase. Sure enough, it's a complex spectrum that is, it has lots of components; try an index of 0 to hear a sine wave, if you're suspicious.

**Akikora**

If you looked at the spectrum of our first example, and compared it to the spectrum Chowning works out, you may wonder what's gone awry. The result can be expressed: You can chew up any amount of free time calculating the resulting side band amplitudes — see the immortal classic: Schottstaedt, "The Simulation of Natural Instrument Tones Using Frequency Modulation with a Complex Modulating Wave". Or use phase modulation instead; in that case, we have effectively oscil gen 0. Or perhaps most forthright, start with the formula for Jn B given above the integral , and say "we want cos sin expanded as a sum of cosines, and we define Jn to be the nth coefficient in that sum". The Bessel functions are nearly 0 until the index B equals the order n. Eventually, the sine slope lessens as it reaches its bottom , the overall phase catches up, and the bouncing stops for that cycle.

**Turisar**

The modulating signal becomes a sum of two or three narrow band noises narrow because normally the amplitude of the noise is low , and these modulate the carrier. Perhaps history can help? Since sine is not linear it is , this is not the same thing as In the second case we just add together the two simple FM spectra, but in the first case we get a more complex mixture involving all the sums and differences of the modulating frequencies. IRE, vol 10, pp , Feb