Spectrograms and the STFT

"A single spectrum tells me what frequencies exist. A spectrogram tells me when they showed up to work, and when they quietly clocked out."

A Time-Aware AI Agent

This section assumes you are comfortable with sampling and the Nyquist rate from Chapter 3 and with windowing and spectral leakage from Section 7.1. Everything here builds directly on the single-frame DFT; the STFT is simply that DFT applied on a moving window.

From one transform to a moving one

The STFT slides a window \(w[n]\) of length \(L\) along the signal \(x[n]\), advancing by a hop \(H\) samples between frames, and computes a DFT of each windowed segment. For frame index \(m\) and frequency bin \(k\):

Three knobs fully determine the transform. The window length \(L\) sets how much signal each frame sees. The hop \(H\) sets how far the window jumps, so the frame overlap is \(1 - H/L\). The window shape \(w[n]\) (Hann, Hamming, Gaussian) controls leakage exactly as in a single-frame FFT. The spectrogram is the magnitude squared, \(S[m,k] = |X[m,k]|^2\), almost always viewed in decibels, \(10\log_{10} S\), because sensor energy spans many orders of magnitude and the log compresses it into something the eye and a network can both use.

Why overlap at all? A window tapers to zero at its edges, so events near a frame boundary are attenuated. Overlapping frames (typically 50% to 75%, that is \(H = L/2\) to \(L/4\)) guarantee every sample is seen near the center of at least one window. The formal requirement for a spectrogram you can invert back to a waveform is the Constant Overlap-Add (COLA) condition: the shifted windows must sum to a constant. A Hann window with 50% overlap satisfies COLA, which is why it is the default almost everywhere.

Choosing parameters so the picture is honest

Start from the physics, not from a default. Ask what the fastest event you care about is and what the closest two frequencies you must separate are. If a bearing impact lasts about 2 ms, a 256 ms window will paint it as a vertical smear with no useful frequency content; you want a window closer to a few milliseconds. If instead you must distinguish a 49.5 Hz line-frequency tone from a 50 Hz one, you need \(\Delta f \le 0.5\) Hz, hence \(L \ge f_s / 0.5\) samples, a window of at least two seconds. These two goals can be flatly incompatible on the same signal, which is exactly why the same chapter also offers wavelets (Section 7.3) that adapt resolution across the band.

Two further points prevent self-deception. First, zero-padding each frame to a longer FFT length makes the spectrogram look smoother but adds no real resolution; it interpolates the same underlying \(\Delta f\). Second, the vertical axis only reaches \(f_s/2\); anything above Nyquist has already aliased into the band before the STFT ever ran, so a clean-looking spectrogram can still be built on corrupt data if the anti-alias filtering from Chapter 6 was skipped.

import numpy as np
from scipy.signal import ShortTimeFFT
from scipy.signal.windows import hann

fs = 4000                       # sample rate, Hz
t = np.arange(0, 2.0, 1/fs)     # two seconds

# A tone that sweeps 200 -> 800 Hz, plus a short impact at t = 1.2 s
sweep = np.sin(2*np.pi*(200 + 300*t)*t)
impact = np.exp(-((t - 1.2)/0.004)**2) * np.sin(2*np.pi*1500*t)
x = sweep + 0.8*impact + 0.05*np.random.randn(t.size)

L, hop = 256, 64                          # 64 ms window, 75% overlap
win = hann(L, sym=False)                  # COLA-friendly taper
SFT = ShortTimeFFT(win, hop=hop, fs=fs, scale_to="magnitude")
Sx = SFT.stft(x)                          # complex, shape (freqs, frames)

power_db = 20*np.log10(np.abs(Sx) + 1e-9)
freqs = SFT.f                             # Hz axis
times = SFT.t(x.size)                     # seconds axis
print(power_db.shape, freqs[-1], round(fs/L, 1), "Hz per bin")

The code above makes the tradeoff tangible: run it, then set L = 64 and re-run. The impact snaps into a crisp column while the sweep's ridge fattens vertically. That single experiment teaches more about the Gabor limit than any inequality.

Reading a spectrogram like an instrument

Once you can build one, learn to read it. Steady tones are horizontal lines; a machine running at constant speed shows a fundamental plus evenly spaced harmonics stacked above it. A rising or falling speed bends those lines into sloped ridges, the basis of engine-order tracking in vehicles. Broadband transients (impacts, footsteps, switch clicks) are vertical streaks that span many frequencies at one instant. Amplitude modulation, the hallmark of a developing bearing fault, shows as sidebands: faint lines flanking a strong carrier, spaced by the modulation rate. Recognizing these four patterns (lines, ridges, streaks, sidebands) turns the spectrogram from a pretty picture into a diagnostic readout, and it is the visual vocabulary that the envelope and cepstral methods of Section 7.5 then quantify.

import librosa
mel = librosa.feature.melspectrogram(y=x, sr=fs, n_fft=256, hop_length=64, n_mels=40)
mel_db = librosa.power_to_db(mel, ref=np.max)

What's Next

In Section 7.3, we lift the fixed-window constraint that the Gabor limit imposes on the STFT. Wavelets and multiresolution analysis use short windows at high frequencies and long windows at low ones, giving sharp timing for transients and fine frequency detail for slow trends in the same transform, exactly the flexibility a single spectrogram cannot offer.