Chapter 4

The power spectral density

Stare at a noise time-trace for as long as you like: it will tell you almost nothing. The question that has an answer is "how much noise power lives at which frequencies?" — and the power spectral density is that answer. Every noise number you will ever quote — V/√Hz on a datasheet, dBc/Hz on a phase-noise plot, pT/√Hz for a magnetometer — is this one object in different clothes. This is the keystone chapter of the course.

From the FFT to the PSD

Chapter 3 left us with random processes: signals $x(t)$ whose individual wiggles are unpredictable but whose statistics are stable. The natural instinct of anyone with an oscilloscope is to record a stretch of the signal for a time $T$ and take its Fourier transform,

$$ X_T(f) \;=\; \int_0^{T} x(t)\, e^{-i 2\pi f t}\, dt . $$

Look at what this integral is: at each instant it takes the sample $x(t)$, points it in the direction $e^{-i2\pi f t}$ (chapter 1's phasor, run backwards at the analysis frequency), and adds it to a running sum. $X_T(f)$ is the endpoint of a walk in the complex plane. That one picture contains the whole definition of the PSD. For a deterministic tone analyzed at its own frequency, every step points the same way — the walk is a straight ray, and $|X_T(f)|$ grows like $T$. Analyzed off-frequency, the steps' directions rotate at the detuning and the walk just circles — no growth: that is why a tone contributes only at its frequency. For noise, the steps have random signs — the walk is exactly chapter 3's random walk, so $|X_T(f)|$ grows only like $\sqrt{T}$ and $|X_T(f)|^2$ like $T$, not $T^2$. Watch all three:

$|X_T(f)|^2$ itself therefore never settles down as you record longer — but $|X_T(f)|^2 / T$ does (in expectation). So we define the power spectral density:

$$ S_x(f) \;=\; \lim_{T\to\infty} \frac{2\,\big\langle |X_T(f)|^2 \big\rangle}{T}, \qquad f \ge 0 . $$

The angle brackets are an ensemble average (over repeated records), and the factor of 2 folds the mathematically symmetric negative-frequency half of the spectrum onto the physical, positive-frequency half. The payoff of this exact normalization is one beautiful identity — Parseval's theorem in its most useful form:

$$ \int_0^{\infty} S_x(f)\, df \;=\; \operatorname{Var}(x) \;=\; x_{\mathrm{rms}}^2 \qquad \text{(zero-mean $x$)} . $$

The area under the PSD is the mean-square value of the signal. Everything in this chapter, and much of the rest of the course, is an application of that sentence. Watch the identity assemble itself: sweep the cutoff below and admit the signal's frequencies one decade at a time.

One more connection, so tidy it deserves its own paragraph. The autocorrelation $R_x(\tau) = \langle x(t)\,x(t+\tau)\rangle$ from chapter 3 and the PSD are a Fourier-transform pair (the Wiener–Khinchin theorem):

$$ S_x(f) \;=\; 2\!\int_{-\infty}^{\infty} R_x(\tau)\, e^{-i 2\pi f \tau}\, d\tau \qquad (f > 0,\ \text{one-sided}). $$

The two descriptions of "memory" are the same object: a delta-function correlation (no memory at all) transforms to a flat, white spectrum, while a slowly decaying correlation (long memory) transforms to a spectrum concentrated at low frequency. Setting $\tau = 0$ recovers the area rule: $R_x(0) = \operatorname{Var}(x) = \int_0^\infty S_x\, df$.

Chapter 3's three autocorrelation stories now line up as one dial. Both panels below are measured from the same record; the theorem is the claim that they always agree with each other:

Units: it is a density

If $x$ is a voltage in volts, then $|X_T|^2/T$ carries units of $\mathrm{V^2\,s} = \mathrm{V^2/Hz}$. So the PSD of a voltage is measured in V²/Hz — volts squared per hertz. The "per hertz" matters in exactly the way "per cubic metre" matters for mass density: the PSD is not a power, it is a power density along the frequency axis. The power in a band is the area over that band, $$ P_{[f_1,f_2]} \;=\; \int_{f_1}^{f_2} S_x(f)\, df , $$ just as mass is density times volume. A PSD value at a single frequency means nothing by itself until you multiply by a bandwidth.

Datasheets usually quote the square root of the PSD, the amplitude spectral density (ASD), $\sqrt{S_x(f)}$ in $\mathrm{V/\sqrt{Hz}}$ — those famous "nV/√Hz" numbers on every op-amp datasheet. Same information, amplitude units instead of power units. To get an rms from an ASD you multiply by the square root of your bandwidth (for a flat spectrum): $x_{\mathrm{rms}} = \sqrt{S_x}\cdot\sqrt{\Delta f}$.

The area under the PSD is the RMS you measure

Time to make the central claim physical. Below is a noise voltage — white noise gently rolled off by a one-pole low-pass (corner $f_c$), sampled at rate $f_s$ for a duration $T$; all three are sliders, so you can also watch what the acquisition itself does (a higher $f_s$ extends the spectrum's reach toward its Nyquist edge $f_s/2$; a longer $T$ smooths the estimate). Follow it with an ideal band-pass from $f_1$ to $f_2$ (in the demo: a brick-wall filter in the FFT domain) and read the output on a true-rms voltmeter. The claim is that the voltmeter reading is exactly the square root of the shaded area under the PSD:

$$ x_{\mathrm{rms}}^{\,\text{in band}} \;=\; \sqrt{\int_{f_1}^{f_2} S_x(f)\, df\,} . $$

Drag the band edges. Watch the shaded area, the filtered trace, and the two readouts — one predicted from the area, one measured from the filtered samples. They track each other everywhere: wide band, narrow band, on the flat part, down the roll-off. This is the whole meaning of a PSD.

Two things worth noticing while you play. First, an equal-looking band on the log axis (say 10–100 Hz vs 100–1000 Hz) does not carry equal power: the area rule is about linear hertz, and a decade higher up contains ten times more hertz. Second, when you narrow the band the filtered trace does not just shrink — it becomes slower and more sinusoidal, ringing near the band's centre frequency. A narrow slice of noise looks like a wobbly sine wave: that observation becomes important when we discuss what a spectrum analyzer actually measures.

Estimating a PSD honestly

Now the uncomfortable part. The definition of $S_x$ contains an ensemble average $\langle\cdot\rangle$ that your single lab record does not have. If you just take one FFT of one record and square it — the raw periodogram — you get an estimate whose expectation is the true PSD but whose fluctuation is enormous: each frequency bin is the sum of the squares of two Gaussian numbers (the real and imaginary parts of $X_T$), i.e. a chi-squared variable with 2 degrees of freedom. Its standard deviation equals its mean. Every bin of a raw periodogram is uncertain by about 100% — and recording longer does not help, because a longer record just gives you more bins, each one still 100% scattered.

The fix is to average. Welch's method: chop the record into $K$ segments, window each segment (more on that below), take the periodogram of each, and average the $K$ periodograms bin by bin. The fluctuation shrinks like $1/\sqrt{K}$; the price is frequency resolution, since each segment is shorter — $\Delta f = f_s/n_{\mathrm{seg}}$. That trade (smoothness vs resolution) is the fundamental bargain of spectrum estimation; there is no free lunch, only a well-chosen $K$.

Lines vs noise floors: the spectrum-analyzer trap

A PSD plot in the lab almost always shows two kinds of feature at once: sharp lines (a 50/60 Hz pickup, a carrier, a calibration tone) sticking out of a smooth noise floor. These two objects have different units, and confusing them is the classic spectrum-analyzer mistake.

A pure sine of amplitude $A$ carries power $A^2/2$ concentrated at one frequency — in the $T\to\infty$ limit its PSD is a delta function, $\frac{A^2}{2}\,\delta(f - f_0)$. In a real measurement with resolution $\Delta f$, all of that power lands in essentially one bin, so the height you read in V²/Hz is about $\frac{A^2/2}{\mathrm{ENBW}}$, where ENBW is the bin's effective width from the window callout above ($1.5\,\Delta f$ for our Hann window) — the height depends on your resolution, not on the signal! Halve $\Delta f$ and the spike doubles in height while its area stays fixed at $A^2/2$. The noise floor does the opposite: it is a genuine density, so its level in V²/Hz stays put while its per-bin power shrinks with the bin width. Try it:

Now the trap. A swept spectrum analyzer displays power per resolution bandwidth, not power per hertz. The unit on its screen is dBm per RBW — dBm is the RF world's absolute power unit, decibels relative to 1 mW ($0\ \mathrm{dBm} = 1$ mW, $-30\ \mathrm{dBm} = 1\ \mu$W), and dBm/Hz is a power density in the same currency. On that display everything is reversed: the line height stays fixed as you change RBW (the line's power all fits inside any RBW), while the noise floor moves — up 10 dB for every 10× wider RBW, because a wider filter collects more noise power. Same physics, the opposite visual behaviour, purely because of what is being plotted.

The essential code

The band-pass filter behind the central demo is worth seeing: a brick-wall filter is three steps in the FFT domain. Zeroing a bin $k$ requires also zeroing its negative-frequency mirror $N-k$, or the inverse transform stops being real.

// brick-wall band-pass: FFT, zero everything outside [f1, f2], inverse FFT
const N = x.length;                          // power of 2, sample rate fs
const re = Float64Array.from(x);
const im = new Float64Array(N);
FFT.fft(re, im);
for (let k = 0; k <= N / 2; k++) {
  const f = k * fs / N;
  if (f < f1 || f > f2) {
    re[k] = im[k] = 0;
    if (k > 0 && k < N / 2) re[N - k] = im[N - k] = 0;   // mirror bin
  }
}
FFT.ifft(re, im);        // re[] is now the filtered time trace
// check: Noise.rms(re)² equals the PSD area between f1 and f2

You now own the tool. The obvious next question is what shapes $S(f)$ actually takes when you point it at real hardware — and it turns out that almost everything in a lab lands on a handful of straight lines on a log–log plot. Those straight lines, and the very different personalities behind them, are the next chapter.

Exercises