FM and PM Modulation Spectrum (interactive)

This lit­tle wid­get will cal­cu­late the spec­trum for var­i­ous mod­u­la­tion sig­nals with arbi­trary fre­quen­cy and ampli­tude and explain why it looks the way it does. It was writ­ten with opti­cal inter­fer­om­e­try in mind, so the mod­u­la­tion sig­nal is defined in terms of the veloc­i­ty of a mov­ing object, which is sim­ple to con­vert to a fre­quen­cy devi­a­tion, see below.

wave­length:
shape:
fre­quen­cy:
ampli­tude:
num­ber of har­mon­ics: Carson’s rule: 0

Warn­ing: Large dataset required for cal­cu­la­tion may lead to unre­spon­sive brows­er. Con­tin­ue any­way

linlog

PSD [a.u.] vs Fre­quen­cy [Hz]
total frac­tion of pow­er plot­ted: 0

veloc­i­ty [m/s] vs Time [a.u.]

Posi­tion [m] vs Time [a.u.]

Derivation

A fre­quen­cy- (or phase-)modulated car­ri­er sig­nal may be writ­ten as

\begin{equation}
E(t) = \frac{1}{2}\Bigl(A_{0}\exp\Bigl[i\omega_0 t + i \varphi(t) \Bigr] + \mathrm{c.c.}\Bigr) \tag{1}
\end{equation}

where $\omega_0$ is the fre­quen­cy of the mod­u­lat­ed car­ri­er and $A_0$ is the car­ri­er ampli­tude which can be com­plex to account for an arbi­trary ini­tial car­ri­er phase. Then $\varphi(t)$ describes the mod­u­la­tion of the phase, and its deriv­a­tive $d_t \varphi(t)$ describes the fre­quen­cy mod­u­la­tion (since fre­quen­cy is just the deriv­a­tive of phase). The “c.c.” stands for the com­plex con­ju­gate of the pre­ced­ing term and ensures that $E(t)$ is always real-val­ued.

A car­ri­er fre­quen­cy oth­er than zero will sim­ply shift the mod­u­lat­ed sig­nal with­in the spec­trum; there­fore, we can assume $\omega_0 = 0$ with­out los­ing much in terms of gen­er­al­i­ty (we will dis­cuss a relat­ed point about detec­tion lat­er). Also, we set $A_0 = 1$ for sim­plic­i­ty and dis­re­gard the “c.c.” term for now (remem­ber­ing that it is there, though). Then we are left with the com­plex pha­sor

\begin{equation}
\underline{E}(t) = \exp\Bigl[i \varphi(t) \Bigr] = \exp\Bigl[i \intop_0^t \omega(\tau) d\tau \Bigr] \tag{2}
\label{basic-pm}\end{equation}

which describes the (same) sig­nal in terms of a phase and (angu­lar) fre­quen­cy mod­u­la­tion, respec­tive­ly. It seems rea­son­able to first look at a sim­ple har­mon­ic mod­u­la­tion sig­nal such as

\begin{equation}
\varphi(t) = \Delta\varphi \cos \bigl(\Omega t + \Phi\bigr) \tag{3}
\label{phase-function}\end{equation}

with ampli­tude $\Delta\varphi$, rep­e­ti­tion fre­quen­cy $\Omega$ and ini­tial phase $\Phi$. The cor­re­spond­ing (angu­lar) fre­quen­cy mod­u­la­tion is

\begin{equation}
d_t\varphi(t) = \omega(t) = - \Omega \Delta\varphi \sin \bigl(\Omega t + \Phi\bigr) = - \Delta\omega \sin \bigl(\Omega t + \Phi\bigr) \tag{4}
\label{frequency-function}\end{equation}

with ampli­tude $\Delta\omega$. For­tu­nate­ly, the expo­nen­tial of a cosine can be expand­ed into the fol­low­ing series

\begin{equation}
\exp \bigl( i z \cos \theta \bigr) = \sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, \exp \bigl(i n \theta \bigr) \tag{5}
\label{jacobi-anger}\end{equation}

It is called the Jaco­bi-Anger expan­sion [1] and the $J_n$ are the $n$th Bessel func­tions of the first kind.
Com­bin­ing (2), (3) and (5) we get

\begin{align}
\underline{E}(t) &= \exp\Bigl[i \Delta\varphi \cos \bigl(\Omega t + \Phi\bigr) \Bigr]\\
&= \exp\Bigl[-i \Delta\omega \intop_0^t \sin \bigl(\Omega \tau + \Phi\bigr) d\tau \Bigr]\\
&= \sum_{n=-\infty}^{\infty} J_n(\Delta\varphi)\, \exp \bigl(i n \pi + \Phi \bigr) \, \exp \bigl(i \Omega t \bigr) \tag{6}
\end{align}

with $i^n = \exp(i n \pi)$. The Bessel coef­fi­cient in the sum gives the mag­ni­tude of each line in the spec­trum, the sec­ond fac­tor gives its phase, and the third fac­tor deter­mines at which fre­quen­cy that line appears in the spec­trum. Hence the lines are neat­ly arranged at mul­ti­ples of the object oscil­la­tion fre­quen­cy.
It appears that the PM and FM spec­trum the­o­ret­i­cal­ly com­pris­es an infi­nite num­ber of spec­tral lines, though the Bessel func­tion becomes very small around $n \approx \Delta\varphi$. In fact, Carson’s rule states that around 98% of the pow­er is con­tained with­in a (two-sided) band­width of [2]

\begin{equation}
BW = 2 \bigl(\Delta\omega_\text{max} + \Omega_\text{max}\bigr) \tag{7}
\label{carsons-rule}\end{equation}

in which the two terms are the max­i­mum fre­quen­cy devi­a­tion and max­i­mum oscil­la­tion fre­quen­cy of the mod­u­la­tion sig­nal. For mod­u­la­tion with a sin­gle cosine this reduces to

\begin{equation}
BW = 2 \Omega\bigl(\Delta\varphi + 1\bigr) \tag{8}
\end{equation}

and basi­cal­ly says to account only for the first $\Delta\varphi + 2$ lines (includ­ing the line at zero fre­quen­cy) which coin­cides well with the above obser­va­tion of Bessel coef­fi­cient mag­ni­tudes. How­ev­er, Carson’s rule real­ly applies only to mod­u­la­tion sig­nals with a con­tin­u­ous spec­trum, or at least one that con­sists of a mul­ti­tude of lines. It is for­tu­itous that it also works for the cosine, and we’ll see below that it doesn’t work always.

Arbitrary Signals

An arbi­trary (peri­od­ic) phase mod­u­la­tion can be Fouri­er-decom­posed into a series of cosines and sines (or cosines with some ini­tial phase), so that

\begin{align}
\underline{E}(t) &= \exp\Bigl[i \sum_{n=0}^\infty \Delta\varphi_n \cos \bigl(n \Omega t + \Phi_n) \Bigr] \\
&= \prod_{n=0}^\infty \exp\Bigl[i \Delta\varphi_n \cos \bigl(n \Omega t + \Phi_n) \Bigr]\tag{9}
\end{align}

The Jaco­bi-Anger expan­sion can then be applied to each fac­tor in the prod­uct sep­a­rate­ly. The spec­trum of the phase-mod­u­lat­ed sig­nal is then obtained by con­vo­lu­tion of the spec­tra of all the fac­tors. Thus, the com­pound spec­trum may be quite dense­ly pop­u­lat­ed over a large band­width when there are many such fac­tors. Here is an exam­ple:

Fig­ure 1: PSD [a.u.] vs Fre­quen­cy [kHz] for a sinu­soidal 100kHz mod­u­la­tion sig­nal with equiv­a­lent veloc­i­ty ampli­tude of 1m/s.


Fig­ure 2: PSD [a.u.] vs Fre­quen­cy [kHz] for a sinu­soidal 90kHz mod­u­la­tion sig­nal with equiv­a­lent veloc­i­ty ampli­tude of 1m/s.


Fig­ure 3: PSD [a.u.] vs Fre­quen­cy [kHz] for a super­po­si­tion of the mod­u­la­tion sig­nals of Fig­ure 1 and Fig­ure 2. Notice that the PSD scale dif­fers from Fig­ures 1 and 2.

Interferometric Units

The val­ues for the inter­ac­tive spec­trum plot at the top of this post are giv­en in inter­fer­o­met­ric units, for we assume that the phase (or fre­quen­cy) mod­u­la­tion is gen­er­at­ed by a mov­ing mir­ror in a sim­ple inter­fer­om­e­ter, e.g. a Michel­son inter­fer­om­e­ter. Our deriva­tion of the spec­trum, how­ev­er, is done in very gen­er­al terms of the mod­u­lat­ed phase $\varphi(t)$ or the mod­u­lat­ed fre­quen­cy $\omega(t)$. Both types of units are relat­ed as

\begin{align}
\varphi(t) &= 2 k z(t) = \frac{4 \pi}{\lambda} z(t) \\
\omega(t) &= d_t \varphi(t) = \frac{4 \pi}{\lambda} d_t z(t) = \frac{4 \pi}{\lambda} v(t)\tag{10}
\end{align}

where $z(t)$ is the (rel­a­tive) posi­tion of the mir­ror, $v(t)$ is its veloc­i­ty, $k$ is the wavenum­ber and $\lamb­da$ is the wave­length of light used in the inter­fer­om­e­ter. The addi­tion­al fac­tor of 2 in the equa­tions results from the change in opti­cal path length being twice the change in mir­ror posi­tion, as the light beam with­in the inter­fer­om­e­ter must trav­el the dis­tance $z(t)$ once towards the mir­ror and then again com­ing back after being reflect­ed.

Carrier Frequency

At the begin­ning we assumed the car­ri­er fre­quen­cy $\omega_0 = 0$ to sim­pli­fy the analy­sis. And look­ing at the sym­met­ric spec­tra in Fig­ures 1–3 we might think the PM/FM sig­nals $\underline{E}(t)$ are real-val­ued and could be detect­ed with e.g. a sim­ple pho­to­di­ode. Look­ing at (6) we can see that this is not the case. While the mag­ni­tudes of the spec­trum of neg­a­tive fre­quen­cies are indeed equal to those of the respec­tive pos­i­tive fre­quen­cies because of

\begin{equation}
J_{-n}(x) = (-1)^n J_n(x)
\end{equation}

for inte­ger $n$, the phas­es do not cor­re­spond to com­plex con­ju­gate pairs. Hence, to prop­er­ly detect such a com­plex sig­nal requires addi­tion­al hard­ware [3]. Alter­na­tive­ly, to prop­er­ly detect a PM/FM sig­nal with just a pho­to­di­ode, we could use a car­ri­er with $\omega_0 \gt 0$. How­ev­er, the fre­quen­cy shift $\omega_0$ would have to be large enough so that the whole two-sided mod­u­la­tion spec­trum of $\underline{E}(t)$ would no longer gen­er­ate any sig­nif­i­cant spec­tral com­po­nents at neg­a­tive fre­quen­cies. If $\omega_0$ were too small, the spec­tra of the two terms in (1) would over­lap, as shown below, caus­ing dis­tor­tions in the detect­ed posi­tion / veloc­i­ty.

Fig­ure 4: PSD [a.u.] vs Fre­quen­cy [kHz] for a real-val­ued FM sig­nal with sinu­soidal 100kHz mod­u­la­tion, equiv­a­lent veloc­i­ty ampli­tude of 1m/s and (suf­fi­cient) car­ri­er fre­quen­cy of 5MHz.

Fig­ure 5: PSD [a.u.] vs Fre­quen­cy [kHz] for a real-val­ued FM sig­nal with sinu­soidal 100kHz mod­u­la­tion, equiv­a­lent veloc­i­ty ampli­tude of 1m/s and (insuf­fi­cient) car­ri­er fre­quen­cy of 2.82MHz.

Fig­ure 6: Demod­u­lat­ed veloc­i­ties for the sig­nals in Fig­ure 4 (blue) and Fig­ure 5 (red).

While PM or FM is some­what resilient to inter­fer­ence due to its spread-spec­trum nature, it is not immune.

References

[1] Jaco­bi-Anger expan­sion [Wikipedia]
[2] Carson’s rule [Wikipedia]
[3] Homo­dyne ver­sus Het­ero­dyne


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