10.3 Short-time Fourier transforms

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §10.3. Rewritten in original wording. Continues 10.2b. Next: 10.3b.

Why chop the timeline

Global sinusoids have constant amplitude for all $t$, so they overlap everywhere and store correlated coefficients — awkward for compression. Lossless compression removes repetition; lossy compression also drops transform pieces consistent with uncertainty and a quality budget (JPEG 2000 uses wavelets).

Think of the FT as an overlap / filter:

$$Y(\omega)=\int\mathrm{d}t\,\frac{e^{-i\omega t}}{\sqrt{2\pi}}\,y(t)=\langle\omega|y\rangle.$$

For Figure 10.1’s staged tones, dissect time into windows centered at $\tau_1,\tau_2,\ldots$ and DFT each piece — a short-time Fourier transform (STFT) sequence $Y^{(\mathrm{ST})}(\omega,\tau)$.

Windowed definition

Slide a window $w(t-\tau)$ (zero outside a chosen support) across the signal and Fourier-transform each chunk. Pick the STFT integral (and the fixed-window tradeoff) among lookalikes.

(Sign convention on the exponential matches the book’s STFT line; keep forward/inverse consistent with Chapter 9 when you code.) Only the chunk under the window contributes. The extra argument $\tau$ is the window center — display as a time–frequency surface (spectrogram). Wavelets improve on fixed width by using scale-dependent windows (narrow for highs, wide for lows) — next sections.

Compression link

Fewer correlated coefficients + discarding small scales → lossy compression. STFT is a historical stepping stone; wavelets systematize multi-resolution packing.

Quiz

Short-time Fourier

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STFT adds which variable beyond a plain FT?

Pick an answer to see if you are right.

Next: 10.3b STFT labs.