Sinusoidal Frequency Modulation (FM)

Frequency Modulation (FM) is well known as the broadcast signal format for FM radio. It is also the basis of the first commercially successful method for digital sound synthesis. Invented by John Chowning [14], it was the method used in the the highly successful Yamaha DX-7 synthesizer, and later the Yamaha OPL chip series, which was used in all ``SoundBlaster compatible'' multimedia sound cards for many years. At the time of this writing, descendants of the OPL chips remain the dominant synthesis technology for ``ring tones'' in cellular telephones.

A general formula for frequency modulation of one sinusoid by another can be written as

$\displaystyle x(t) = A_c\cos[\omega_c t + \phi_c + A_m\sin(\omega_m t + \phi_m)], \protect$

(4.5)

where the parameters $(A_c,\omega_c,\phi_c)$ describe the carrier sinusoid, while the parameters $(A_m,\omega_m,\phi_m)$ specify the modulator sinusoid. Note that, strictly speaking, it is not the frequency of the carrier that is modulated sinusoidally, but rather the instantaneous phase of the carrier. Therefore, phase modulation would be a better term (which is in fact used). Potential confusion aside, any modulation of phase implies a modulation of frequency, and vice versa, since the instantaneous frequency is always defined as the time-derivative of the instantaneous phase. In this book, only phase modulation will be considered, and we will call it FM, following common practice.^4.8 Figure 4.14 shows a unit generator patch diagram [42] for brass-like FM synthesis. For brass-like sounds, the modulation amount increases with the amplitude of the signal. In the patch, note that the amplitude envelope for the carrier oscillator is scaled and also used to control amplitude of the modulating oscillator. figure[htbp] $\includegraphics{eps/fmug}$ It is well known that sinusoidal frequency-modulation of a sinusoid creates sinusoidal components that are uniformly spaced in frequency by multiples of the modulation frequency, with amplitudes given by the Bessel functions of the first kind [14]. As a special case, frequency-modulation of a sinusoid by itself generates a harmonic spectrum in which the

th harmonic amplitude is proportional to $J_k(\beta)$ , where

is the order of the Bessel function and $\beta$ is the FM index. We will derive this in the next section.^4.9

Bessel Functions

The Bessel functions of the first kind may be defined as the coefficients $J_k(\beta)$ in the two-sided Laurent expansion of the so-called generating function [84, p. 14],^4.10

$\displaystyle e^{\frac{1}{2}\beta\left(z-\frac{1}{z}\right)} = \sum_{k=-\infty}^\infty J_k(\beta) z^k \protect$

(4.6)

where

is the integer order of the Bessel function, and $\beta$ is its argument (which can be complex, but we will only consider real $\beta$ ). Setting $z=e^{j\omega_mt}$ , where $\omega_m$ will interpreted as the FM modulation frequency and

as time in seconds, we obtain

$\displaystyle x_m(t)\isdef e^{j\beta\sin(\omega_m t)} = \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}. \protect$

(4.7)

The last expression can be interpreted as the Fourier superposition of the sinusoidal harmonics of $\exp[j\beta\sin(\omega_m t)]$ , i.e., an inverse Fourier series sum. In other words, $J_k(\beta)$ is the amplitude of the

th harmonic in the Fourier-series expansion of the periodic signal

. Note that $J_k(\beta)$ is real when $\beta$ is real. This can be seen by viewing Eq. $\,$ (4.6) as the product of the series expansion for $\exp[(\beta/2) z]$ times that for $\exp[-(\beta/2)/z]$ (see footnote pertaining to Eq. $\,$ (4.6)). Figure 4.15 illustrates the first eleven Bessel functions of the first kind for arguments up to $\beta=30$ . It can be seen in the figure that when the FM index $\beta$ is zero,

and

for all

. Since $J_0(\beta)$ is the amplitude of the carrier frequency, there are no side bands when $\beta=0$ . As the FM index increases, the sidebands begin to grow while the carrier term diminishes. This is how FM synthesis produces an expanded, brighter bandwidth as the FM index is increased.

**Figure 4.15:** Bessel functions of the first kind for a range of orders and argument $\beta$ .
$\includegraphics[width=\twidth]{eps/bessel}$

FM Spectra

Using the expansion in Eq. $\,$ (4.7), it is now easy to determine the spectrum of sinusoidal FM. Eliminating scaling and phase offsets for simplicity in Eq. $\,$ (4.5) yields

$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$

(4.8)

where we have changed the modulator amplitude

to the more traditional symbol $\beta$ , called the FM index in FM sound synthesis contexts. Using phasor analysis (where phasors are defined below in §4.3.11),^4.11i.e., expressing a real-valued FM signal as the real part of a more analytically tractable complex-valued FM signal, we obtain

$\displaystyle x(t) \isdef \cos[\omega_c t + \beta\sin(\omega_m t)]$	$\displaystyle =$	re $\displaystyle \left\{e^{j[\omega_c t + \beta\sin(\omega_m t)]}\right\}$
	$\displaystyle =$	re $\displaystyle \left\{e^{j\omega_c t} e^{j\beta\sin(\omega_m t)}\right\}$
	$\displaystyle =$	re $\displaystyle \left\{e^{j\omega_c t} \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}\right\}$
	$\displaystyle =$	re $\displaystyle \left\{\sum_{k=-\infty}^\infty J_k(\beta) e^{j(\omega_c+k\omega_m) t}\right\}$
	$\displaystyle =$	$\displaystyle \sum_{k=-\infty}^\infty J_k(\beta) \cos[(\omega_c+k\omega_m) t]$	(4.9)

where we used the fact that $J_k(\beta)$ is real when $\beta$ is real. We can now see clearly that the sinusoidal FM spectrum consists of an infinite number of side-bands about the carrier frequency $\omega_c$ (when $\beta\neq 0$ ). The side bands occur at multiples of the modulating frequency $\omega_m$ away from the carrier frequency $\omega_c$ .

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Analytic Signals and Hilbert Transform Filters
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Sinusoidal Amplitude Modulation (AM)