Spectral Analysis

Why Does This Exist?

A measured trace mixes tones, broadband noise, and interference; decisions (bearing fault? grid oscillation? seizure onset?) need power per frequency with uncertainty attached. The raw periodogram (squared DFT magnitudes) is unbiased but inconsistent: its variance never shrinks with more data, so longer records merely add wobblier bins. Spectral analysis fixes estimation: windowed, averaged, and tapered variants that trade resolution for stability with stated degrees of freedom. Transforms are Fourier; time-localized alternatives are wavelets.

The one idea you need first

The power spectral density $S(f)$ gives power per Hz: tone power concentrates in lines, white noise spreads flat at $\sigma^2/f_s$ per bin. Estimates $\hat{S}$ report power per bin with chi-square fluctuations around $S$.

Think of It Like This

How It Actually Works

Periodogram: $\hat{S}_P(k) = |X_k|^2 / (f_s n)$, variance $\approx S^2$ per bin regardless of $n$ (2 degrees of freedom). Bartlett/Welch: split into $M$ segments (Welch: overlapping, windowed), average: variance $/M$, resolution $\times M$ coarser, degrees of freedom $2M$. Multitaper: $K$ orthogonal Slepian tapers on the full record: $2K$ degrees of freedom at full resolution minus bandwidth $W$ widening. Rule: $M \ge 8$ segments (or $K \ge 5$ tapers) before quoting levels.

Worked example

Bearing vibration at $f_s = 12{,}800$ Hz, $n = 12{,}800$ samples ($1$ s): fault tone $320$ Hz at power $4.0$ ($\text{mm/s})^2$, noise floor $0.04$ per bin (ratio $100$: $20$ dB). Raw periodogram bins fluctuate with standard deviation equal to the mean: noise bins read $0.0$ to $0.2$ (some crossing $10\%$ of the tone); the tone bin reads $0.5$ to $12$ across realizations. Welch with $M = 8$ half-overlapping Hann segments: noise standard deviation falls to $0.04/\sqrt{8} \approx 0.014$; tone resolves at $4.0 \pm 1.4$ with $16$ degrees of freedom; $95\%$ interval $[2.2, 7.5]$ (chi-square): detection margin $2.2/0.068 \approx 32$ above the noise ceiling. Decision-grade where raw was not.

Code

1import cmath2import math3
4
5def periodogram(xs: list[float], fs: float) -> list[float]:6    n = len(xs)  # squared magnitudes scaled to power per bin7    out = []8    for k in range(n // 2 + 1):9        s = sum(x * cmath.exp(-2j * cmath.pi * k * t / n) for t, x in enumerate(xs))10        out.append(round(abs(s) ** 2 / (fs * n), 4))11    return out12
13
14tone = [2 * math.cos(2 * math.pi * 2 * t / 16) for t in range(16)]15spec = periodogram(tone, 16.0)16print(spec.index(max(spec)), round(max(spec), 2))  # -> bin 2, power 4.0

Tone power $4.0$ lands in bin $2$ exactly (frequency $2$ Hz at $1$ s): the estimator finds the line where planted.

Watch Out For

The Quick Version

• Power spectra report power per frequency with chi-square fluctuations per bin.
• Raw periodograms never steady with record length: variance equals the mean squared.
• Welch with $8$ segments cuts noise deviation to $0.014$ and gives $16$ degrees of freedom.
• The fault tone at $4.0$ stands $32$ times above the noise ceiling after averaging.
• Multitapers keep resolution while buying stability through orthogonal windows.
• Average before alerting; window lightly when tones sit close.

What to Read Next

• Descriptive Statistics
• Probability Distributions
• Vectors, Matrices and Tensors