## Introduction

In AM modulation, the amplitude of the carrier wave is varied in accordance with the modulation wave. This can be described by the modulation equation:

AM = Ac*[1+m*cos(wm*t)]*cos(wc*t)]

Modulation Waveform = Am*cos(wm*t)

Carrier Wave = Ac*cos(wc*t)

m = modulation index = Am/Ac <=1

When m = 1 (100%), the amplitude of the mod waveform is equal to the carrier amplitude and this results in the deepest modulation where the mod carrier reaches 2xAc.

cos(wm*t)*cos(wc*t) = 1/2[cos((wm+wc)*t)) + cos((wc_wm)*t)]

Since the product of two cosine waves is equal to one half the sum of the cosine of the sum of frequencies plus the cosine of the difference of frequencies, the spectrum of the modulation has a USB and LSB component at (wc+wm) & (wc-wm). At 100% modulation, the sidebands are down by 10*log(1/4) = -6dB. At 50% modulation, the sidebands are down by 10*log(m^2/4)) = 10*log(1/16) = -12dB.

## AM Modulation Schematic on GNU Radio

Figure 1 shows the GNU Radio schematic used to generate the AM modulation. Two signal sources are used, one for the mod waveform and one for the carrier. The sources are set to float for simplicity. Then add & multiply blocks are used to literally build the mod equation. Again all blocks are set to float. Range variables are setup to vary the mod frequency from 300 – 2700Hz, the carrier from 5KHhz to 15KHz and m from 0 to 1.0.

Figure 2 shows m=1 or 100% modulation. The carrier amplitude peaks at 2*Ac and goes down to 0 (Note when m>1, over modulation occurs and this produces severe non linearity). You can see the shape of the lower frequency modulation waveform in the amplitude of the carrier. Figure 3 shows the spectrum, note the USB at 10KHz + 1KHz = 11KHz and the LSB at 10KHz – 1KHz = 9KHz. The level difference is (-14.5dB – (-20.5))= 6dB.

Figure 4 shows m=0.5 or 50% modulation. The carrier amplitude peaks at 1.5*Ac and goes down to 0.5Ac. You can see the shape of the lower frequency modulation waveform in the amplitude of the carrier. Figure 5 shows the spectrum, note the USB at 10KHz + 1KHz = 11KHz and the LSB at 10KHz – 1KHz = 9KHz. The level difference is (-14.5dB – (-26.5))= 12dB.

## References

#1. – “GNU Radio”

https://www.gnuradio.org/