Chapter 1

The pure tone

Everything in this course — phase noise, power spectral densities, Allan deviations, LIGO's sensitivity curve — is a story about one equation with three knobs. This chapter is about the equation itself, so that when we start adding noise to the knobs, nothing else will be new.

A signal with three knobs

Take the most boring signal in physics: a voltage (or a laser field, a trapped ion's drive, the output of a quartz crystal, anything that oscillates) doing exactly the same thing forever,

$$ V(t) = A \cos\!\big(2\pi \nu_0 t + \varphi\big). $$

There are only three things you can adjust:

• the amplitude $A$ — how big the swing is (volts, tesla, watts^1/2… whatever your signal is measured in);
• the frequency $\nu_0$ — how many full cycles happen per second, in hertz;
• the phase $\varphi$ — where in its cycle the oscillation is at $t=0$, in radians.

Drag each slider below and watch which aspect of the curve it controls. The faint curve is a frozen reference ($A=1$, $\nu_0=2\,$Hz, $\varphi=0$) so you can see what changed.

The phasor: one picture instead of trigonometry

Cosines are awkward to manipulate. The clean way to think about an oscillation — and the picture that will carry us through modulation and noise — is a rotating arrow. Euler's formula says

$$ V(t) = A\cos(2\pi\nu_0 t + \varphi) = \mathrm{Re}\Big[\, A\, e^{\,i(2\pi\nu_0 t + \varphi)} \Big], $$

so a pure tone is the shadow (real part) of an arrow of fixed length $A$ rotating counter-clockwise in the complex plane at $\nu_0$ turns per second, having started at angle $\varphi$. Press play:

The angle of the arrow at time $t$,

$$ \Phi(t) = 2\pi\nu_0 t + \varphi, $$

is called the instantaneous phase (also "total phase"). It is the single most important quantity in this course. Notice what the two other knobs are in terms of it: the phase $\varphi$ is just $\Phi(0)$, and the frequency is how fast the phase advances:

$$ \nu(t) \;=\; \frac{1}{2\pi}\,\frac{d\Phi}{dt}. $$

For a pure tone this derivative is the constant $\nu_0$. But keep the formula: frequency is the slope of phase. When the phase starts wandering randomly in chapter 6, this one line is what will connect "phase noise" and "frequency noise" — they are the same wandering, looked at either as position or as velocity.

What a pure tone looks like on a spectrum analyzer

The second home of every signal is the frequency domain. For a truly eternal cosine, all the power sits at exactly one frequency: the spectrum is an infinitely narrow spike at $\nu_0$. But you never observe a signal forever — you record for some time $T$ and ask your FFT (or your spectrum analyzer) what it saw. The result is a peak of finite width

$$ \Delta\nu \sim \frac{1}{T}, $$

the Fourier uncertainty relation: to know a frequency to 1 Hz you must watch for about a second; to know it to 1 mHz, about $10^3\,$s. This width has nothing to do with noise — it is the resolution limit of a finite observation. Try it:

Two tones: the simplest interesting signal

Before we modulate anything, add just one more cosine. Two equal tones at nearby frequencies $\nu_1$ and $\nu_2$ give, by a trigonometric identity,

$$ \cos(2\pi\nu_1 t) + \cos(2\pi\nu_2 t) = 2\, \underbrace{\cos\!\big(2\pi \tfrac{\nu_1-\nu_2}{2}\, t\big)}_{\text{slow envelope}} \;\underbrace{\cos\!\big(2\pi \tfrac{\nu_1+\nu_2}{2}\, t\big)}_{\text{fast carrier}} . $$

In the time domain you see beats: a fast oscillation at the average frequency whose amplitude swells and shrinks at the difference frequency. In the frequency domain you see the truth with no trigonometry at all: just two spikes.

This is worth internalizing now, because it is the pattern of everything to come: a signal that looks complicated in one domain is often trivially simple in the other. A wobbling envelope in time = two spectral lines. And when we modulate a carrier in the next chapter, the modulation will show up as sidebands — extra spikes placed symmetrically around the carrier, exactly like these two.

The same thing in code

Every plot on this page is generated live by a few lines of JavaScript. Here is the essential part (the full source is in this page — view source), in case reading code is how the mathematics clicks for you:

// a pure tone, sampled at fs for T seconds
const fs = 200, T = 4, N = Math.round(fs * T);
const t = [], V = [];
for (let i = 0; i < N; i++) {
  t.push(i / fs);
  V.push(A * Math.cos(2 * Math.PI * nu0 * t[i] + phi));
}

// its spectrum: |FFT|^2, scaled to a power spectral density
const { f, psd } = FFT.periodogram(V, fs);   // chapter 4 explains the scaling

Exercises