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.

#### 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:

#### 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.

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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