Editing Frequency modulation synthesis (section)

== Spectral analysis ==
[[File:DemoFMfreq.wav|thumb|2-operator demonstration: if the frequency of the modulator is lower than that of the carrier, the output note will be that of the modulator.]]
There are multiple variations of FM synthesis, including:
*Various operator arrangements (known as "FM Algorithms" in Yamaha terminology)
**2 operators
**Serial FM (multiple stages)
**Parallel FM (multiple modulators, multiple-carriers),
**Mix of them
*Various waveform of operators
**Sinusoidal waveform
**Other waveforms
*Additional modulation
**Linear FM
**Exponential FM (preceded by the [[anti-logarithm]] conversion for CV/oct. interface of analog synthesizers)
**[[Oscillator sync]] with FM
''etc''.

As the basic of these variations, we analyze the spectrum of 2 operators (linear FM synthesis using two sinusoidal operators) on the following.

=== 2 operators ===
The spectrum generated by FM synthesis with one modulator is expressed as follows:<ref>{{harvnb|Chowning|1973|pp=1–2}}</ref><ref>
{{cite web
 | last      = Doering | first = Ed
 | title     = Frequency Modulation Mathematics
 | url       = http://cnx.org/content/m15482/latest/
 | access-date =  2013-04-11
 }}</ref>

For modulation signal <math>m(t) = B\,\sin(\omega_m t)\,</math>, the carrier signal is:<ref group="note">Note that modulation signal <math>m(t)</math> as [[instantaneous frequency]] is converted to the [[phase (waves)|phase]] of carrier signal <math>FM(t)</math>, by time integral between <math>[0, t]</math>.</ref>
:<math>\begin{align}
FM(t)	& \ =\ A\,\sin\left(\,\int_0^t \left(\omega_c + B\,\sin(\omega_m\,\tau)\right)d\tau\right) \\
	& \ =\ A\,\sin\left(\omega_c\,t - \frac{B}{\omega_m}\left(\cos(\omega_m\,t) - 1\right)\right) \\
	& \ =\ A\,\sin\left(\omega_c\,t + \frac{B}{\omega_m}\left(\sin(\omega_m\,t - \pi/2) + 1\right)\right) \\
\end{align}</math>

If we were to ignore the constant phase terms on the carrier <math>\phi_c = B/\omega_m\,</math> and the modulator <math>\phi_m = - \pi/2\,</math>, finally we would get the following expression, as seen on {{harvnb|Chowning|1973}} and {{harvnb|Roads|1996|p=[https://books.google.com/books?id=nZ-TetwzVcIC&&pg=PA232 232]}}:
:<math>\begin{align}
FM(t)	& \ \approx\ A\,\sin\left(\omega_c\,t + \beta\,\sin(\omega_m\,t)\right) \\
	& \ =\ A\left( J_0(\beta) \sin(\omega_c\,t)
	     + \sum_{n=1}^{\infty} J_n(\beta)\left[\,\sin((\omega_c+n\,\omega_m)\,t)\ +\ (-1)^{n}\sin((\omega_c-n\,\omega_m)\,t)\,\right] \right) \\
	& \ =\ A\sum_{n=-\infty}^{\infty} J_n(\beta)\,\sin((\omega_c+n\,\omega_m)\,t)
\end{align}</math>
where <math>\omega_c\,,\,\omega_m\,</math> are [[angular frequency|angular frequencies]] (<math>\,\omega = 2\pi f\,</math>) of carrier and modulator, <math>\beta = B / \omega_m\,</math> is [[frequency modulation#Modulation index|frequency modulation index]], and [[amplitude]]s <math>J_n(\beta)\,</math> is <math>n\,</math>-th [[Bessel function#Bessel functions of the first kind : Jα|Bessel function of first kind]], respectively.<ref group="note">The above expression is transformed using [[List of trigonometric identities#Angle sum and difference identities|trigonometric addition formulas]]
: <math>\begin{align}
	\sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y
    \end{align}</math>
and a lemma of Bessel function

: <math>\begin{align}
	\cos(\beta\sin \theta) & = J_0(\beta) + 2\sum_{n=1}^{\infty}J_{2n}(\beta)\cos(2n\theta) \\
	\sin(\beta\sin \theta) & = 2\sum_{n=0}^{\infty}J_{2n+1}(\beta)\sin((2n+1)\theta)
    \end{align}</math>
: ('''Source''': {{harvnb|Kreh|2012}})
as following:

: <math>\begin{align}
	& \sin\left(\theta_c + \beta\,\sin(\theta_m)\right) \\
	& \ =\ \sin(\theta_c)\cos(\beta\sin(\theta_m)) + \cos(\theta_c)\sin(\beta\sin(\theta_m)) \\
	& \ =\ \sin(\theta_c)\left[J_0(\beta) + 2\sum_{n=1}^{\infty}J_{2n}(\beta)\cos(2n \theta_m)\right] 
	     + \cos(\theta_c)\left[2\sum_{n=0}^{\infty}J_{2n+1}(\beta)\sin((2n+1)\theta_m)\right] \\
	& \ =\ J_0(\beta) \sin(\theta_c)
        +  J_1(\beta) 2\cos(\theta_c)\sin(\theta_m)
    	+  J_2(\beta) 2\sin(\theta_c)\cos(2\theta_m)
    	+  J_3(\beta) 2\cos(\theta_c)\sin(3\theta_m)
        +  ... \\
	& \ =\ J_0(\beta) \sin(\theta_c)
	     + \sum_{n=1}^{\infty} J_n(\beta)\left[\,\sin(\theta_c + n\theta_m)\ +\ (-1)^{n}\sin(\theta_c - n\theta_m)\,\right] \\
	& \ =\ \sum_{n=-\infty}^{\infty} J_n(\beta)\,\sin(\theta_c + n\theta_m)\qquad(\because\ J_{-n}(x) = (-1)^n J_{n}(x))
    \end{align}</math></ref>