FM Synthesis: Formula & Spectrum

FM Equation¶

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

$$ x(t) = A \sin(2 \pi f_c t + I \sin(2 \pi f_m t) ) $$

The modulation index $I$ determins the strength of the modulation effect, relative to the modulator frequency. It can be expressed as:

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

Spectrum of Frequency Modulation¶

Compared to amplitude modulation techniques, FM generates more - as in infinite - spectral components, which can be illustrated when calculating the Fourier transform of the FM formula:

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

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

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

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

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

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

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

Bessel functions¶

The Fourier transform of the equation above can be expressed through a series of delta impulses, with amplitudes being derived from Bessel functions $J_n$:

$$X(f) = \sum_{n = -\infty}^{\infty} J_n^2(\beta) \delta(f - f_c - n f_m)$$

Since this series is infinite, the spectrum of FM signals has an infinite number of sidebands which are increased in energy for high modulation indices.

Bessel Functions of first Kind¶

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.