# FM Synthesis: Formula & Spectrum

### FM Formula¶

Frequency modulation with two sinusoidal oscillators can be written as follows:

$$x(t) = A \sin(\omega_\alpha t + I \sin(\omega_\beta t) )$$

Modulation Index:

$$I = \frac{\Delta f}{\Delta f_m}$$

### Spectrum of Frequency Modulation¶

Compared to amplitude modulation techniques, FM generates more spectral components, which can be illustrated when calculating the Fourier transform of the FM formula.

Based on the FM formula, the spectrum can be calculated using trigonometric identities:

$$x(t) = \cos(\omega_\alpha t + I \sin(\omega_\beta t) )$$

$\text{with:} \cos(a+b) = \cos(a) \cos(b) - \sin(a) \sin(b)$

$$x(t) = \cos(\omega_\alpha t) \cos(I \sin(\omega_\beta t)) - \sin(\omega_\alpha t) \sin(I \sin(\omega_\beta t))$$

$\text{with:} \cos(a) \cos(b) = \frac{1}{2} \left( \cos(a-b) + \cos(a+b) \right)$

$\text{and:} \sin(a) \sin(b) = \frac{1}{2} \left( \cos(a-b) - \cos(a+b) \right)$

$$\begin{eqnarray} &=& \frac{1}{2} ( \sin(\omega_\alpha t + I \sin(\omega_\alpha t)) \\ &&+ \sin(\omega_\alpha t - I \sin(\omega_\alpha t))\\ &&+ \sin(\omega_\alpha t + I \sin(\omega_\alpha t)) \\ &&+ \sin( I \sin(\omega_\alpha t) - \omega_\alpha t) ) \end{eqnarray}$$

Bessel functions

The above equation can be expressed through Bessel functions:

$$\begin{eqnarray} \sin(\omega_\alpha t + I \sin(\omega_\alpha t)) &=& J_0(I) \cos(\omega_\alpha) \\ % && + J_1(I) \cos(\omega_\alpha - \omega_\beta)t - \cos(\omega_\alpha + \omega_\beta)t \\ % && - J_2(I) \cos(\omega_\alpha - 2 \omega_\beta)t + \cos(\omega_\alpha + 2 \omega_\beta)t \\ % && + ... \end{eqnarray}$$

The spectrum of FM signals thus has an infinite number of sidebands which are increased in energy for high modulation indices.

### Harmonic vs Inharmonic¶

Depending on the ratio between modulator and carrier, the FM spectrum has different properties. In the formula for the FM sidebands is is obvious that for integer ratios between carrier to modulator, all sidebands are integer multiples of the fundamental frequency.

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