Daqarta
Data AcQuisition And Real-Time Analysis
Scope - Spectrum - Spectrogram - Signal Generator
Software for Windows
The following is from the Daqarta Help system:

# AM Depth

Macro: AMdepth

The AM modulation depth can be set from 0 to 200%. Daqarta uses a special scaling scheme for AM. Conventional amplitude modulation would multiply the main wave (carrier) by the sum of one plus the depth-scaled modulator wave:

```
sin(carrier) * [1 + Depth * sin(modulator)]
```

Under that scheme, 0% Depth gives the unmodulated carrier, but 100% Depth gives a modulated wave with peaks twice as large as the unmodulated carrier. That would require that the carrier be readjusted to prevent overdriving the output.

Instead, Daqarta uses a slightly different scheme that gives identical waveforms, but automatically takes care of scaling to insure that the modulated wave exactly fills the available range:

```
sin(carrier) * [1 - Depth/2 + Depth/2 * sin(modulator)]
```

200% Depth is equivalent to what is sometimes called "Ring Modulation" by those familiar with old analog music synthesis, named after an early circuit that used 4 diodes connected head-to-tail in a (square) ring to perform multiplication. The above scheme reduces to simple multiplication of the two sine waves in this case. This is used in the Sine Wave Multiplication Experiment that demonstrates the heart of Fourier analysis. It's also used in the Monaural Beats discussion of the Monaural and Binaural Beats mini-app.

It is very instructive to view the spectrum of an AM sine wave while you adjust depth and modulation frequency. With Depth set to 0, you see a single line in the spectrum at the main Freq value. (Use the Spectrum Window function to get rid of distracting spectral leakage skirts, or better yet use the Step Lines option to set the main Freq to land exactly on a spectral line.)

As you start to raise Depth, you will notice two flanking lines around the main one, each at a distance equal to the modulator frequency. They will rise as Depth increases until at 100% they are half the magnitude of the main line. Keep going, and they keep rising; at 133.33% all three components will be the same size, but above that the main carrier becomes smaller than the sidebands, until finally at 200% (simple multiplication) only the sidebands remain... no carrier at all! (Also known as "Suppressed Carrier Modulation".)

This is exactly as predicted by the sine multiplication formula from high school trigonometry:

```
sin(A) * sin(B) = 1/2 * cos(A-B) - 1/2 * cos(A+B)
```

All you have are sum and difference frequencies in the product; neither of the original frequencies are present. Note that this is not limited to sine wave modulators; whatever the modulator spectrum, it will be mirrored about the carrier frequency. (Essentially, you apply the above formula to each modulator component separately.)

This is a powerful tool. If you use a low-pass noise for the modulator, you will get a band of noise twice that width centered about the carrier. You can then apply FM or a frequency sweep to the carrier, and it becomes a noise band with a changing center frequency.

For pure sinusoids, the sideband amplitudes are always 1/4 of AM Depth. Note that the modulated AM carrier amplitude plus the two sideband amplitudes will always sum to unity (assuming carrier Level = 100%). For example, if AM Depth = 128% then each sideband amplitude is 128 / 4 = 32% and the carrier is 100 - 2 * 32 = 36 percent.

```
Each Sideband Amplitude = (AM Depth) / 4

Carrier Amplitude = 100% - (AM Depth) / 2
```

NOTE: AM Depth allows entry of negative values, which give the equivalent of inverted (180 degree) modulator phase.

One powerful application of AM is in conjunction with Stream Modulation for multiple tone creation.

Macro Notes:

L.1.AMdepth=50 sets Left Stream 1 AM Depth to 50%.

L.1.AMdepth=>1 increments AM Depth by 1%, while L.1.AMdepth=>-1 decrements by 1%. Only +/-1% steps are accepted.

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