# AM & Ringmodulation: Formula & Spectrum

## Amplitude Modulation vs Ringmodulation¶

Both amplitude modulation and ringmodulation are a multiplication of two signals. The basic formula is the same for both:

$y[n] = x[n] \cdot m[n]$

However, for ringmodulation the modulation signal is symmetric:

$y[n] = \sin\left(2 \pi f_c \frac{n}{f_s}\right) \cdot \left(\sin\left[2 \pi f_m \frac{n}{f_s}\right]\right)$

Whereas for amplitude modulation, the signal ist asymetric:

$y[n] = \sin\left(2 \pi f_c \frac{n}{f_s}\right) \cdot \left( 1+ \sin\left[2 \pi f_m \frac{n}{f_s}\right]\right)$

This differnce has an influence on the resulting spectrum and on the sound, as the following examples show.

### AM Spectrum¶

The spectrum for amplitude modulation can be calculated as follows:

$Y[k] = DFT(y[n])$

$\displaystyle Y[k] = \sum_{n=0}^{N-1} y[n] \cdot e^{-j 2 \pi k \frac{n}{N}}$

$\displaystyle = \sum_{n=0}^{N-1} \sin\left(2 \pi f_c \frac{n}{f_s}\right) \cdot \left( 1+ \sin\left[2 \pi f_m \frac{n}{f_s}\right]\right) \cdot e^{-j 2 \pi k \frac{n}{N}}$

$\displaystyle =\sum_{n=0}^{N-1} \left( \sin\left(2 \pi f_c \frac{n}{f_s}\right) + 0.5 \left( \cos\left(2 \pi (f_c - f_m)\frac{n}{f_s}\right) - \cos\left(2 \pi (f_1 + f_m)\frac{n}{f_s}\right) \right) \right) \cdot e^{-j 2 \pi k \frac{n}{N}}$

$\displaystyle= \delta[f_1] + 0.5 \delta[f_c - f_m] + 0.5 \ \delta[f_c + f_m]$

AM creates a spectrum with a peak at the carrier frequency and two peaks below and above it. Their position is defined by the difference between carrier and modulator.

### Ringmod Spectrum¶

$\mathcal{F} [ y(t)] = \int\limits_{-\inf}^{\inf} y(t) e^{-j 2 \pi f t} \mathrm{d}t$

$= \int\limits_{-\inf}^{\inf} \left( \sin(2 \pi f_c t) \sin(2 \pi f_s t) \right) e^{-j 2 \pi f t} \mathrm{d}t$

$= \frac{1}{2 j} \int\limits_{-\inf}^{\inf} \left( (-e^{-j 2 \pi f_c t} +e^{j 2 \pi f_c t}) (-e^{-j 2 \pi f_s t} +e^{j 2 \pi f_s t}) \right) \ e^{-j 2 \pi f t} \mathrm{d}t$

$= \frac{1}{2 j} \int\limits_{-\inf}^{\inf} \left( e^{j 2 \pi (f_c+f_s) t} - e^{j 2 \pi (f_c-f_s) t} - e^{j 2 \pi (-f_c+f_s) t} + e^{j 2 \pi (-f_c-f_s) t} \right) e^{-j 2 \pi f t}$

$= \frac{1}{2 j} \left[ \delta(f_c+f_s) -\delta(f_c-f_s) - \delta(-f_c+f_s) + \delta(-f_c-f_s) \right]$

Ringmodulation creates a spectrum with
two peaks below and above the carrier frequency. Their position is defined by the difference between carrier and modulator.
The modulator is supressed, since it is symmetric.

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