Spectral features inherit the frame, window, FFT, and normalization contract

The analysis window chooses how many observed samples contribute to one transform. The hop chooses how far consecutive frames advance, while the FFT length can include zero padding beyond the observed window. For real audio, an even transform length commonly yields half that length plus one one-sided complex bins. Count frames under an explicit boundary rule: full frames only, centered frames with padding, and a flushed partial final frame produce different totals. The SciPy 1.16.0 ShortTimeFFT API documents centered columns and its own border conventions; the simpler left-aligned formula here is deliberately stated separately.

Window duration and hop duration are different latencies. A left-aligned frame cannot be fully available until its last required sample arrives; a centered feature referenced to its midpoint needs future samples relative to that timestamp. Later encoder lookahead adds another requirement. Keep feature time, capture time, availability time, and recognition emission time separate. Merely shifting reported timestamps does not turn a future-dependent feature into a causal one, and inserting zeros at a chunk edge is not the same as observing the missing future waveform.

The bin spacing is sample rate divided by FFT length. Increasing the FFT length with zeros samples the same finite-window spectrum more densely; it does not extend the observation or create the ability to separate arbitrarily close tones. Window shape controls leakage and main-lobe width, trading different spectral properties. A short window can localize rapid changes while providing less frequency discrimination. Choose it for the task and training recipe rather than maximizing the number of bins and interpreting that larger matrix as strictly more measured information.

Raw FFT magnitudes scale with window gain and transform convention. For an isolated coherent real sinusoid at an interior bin, a suitable amplitude estimate doubles the positive-frequency magnitude and divides by window sum, provided the negative-frequency image does not contaminate that bin. DC and Nyquist bins use different doubling rules. Power and power spectral density require other normalizations, involving squared gain or window energy and frequency units. Log features also need a declared floor and reference; a magnitude in arbitrary units is not automatically calibrated sound pressure or loudness.

Test impulses, silence, bin-centered and off-bin tones, integer and floating representations, and signals split at awkward chunk sizes. Compare the actual feature tensors, not only final transcriptions. Check window samples, pre-emphasis if used, log base, floors, mel mapping, normalization statistics, and channel mixing under pinned versions. A model can tolerate some differences while failing badly on others; tolerance is an empirical property, not permission to omit preprocessing details. Preserve the mapping from feature frames back to source sample indices so later latency and alignment measurements remain interpretable.

The spectral lab demonstrates a finite harmonic expansion rather than an audio STFT. Reset, explicitly set BOTH mode count N=4 and decay p=1, predict four modeled modes, then Play or Step. Reset and re-set BOTH N=8 and p=1; predict eight. Edits retain t. The analytic reconstruction adds harmonics under its fixed rule; it does not window speech, perform an FFT, or model zero padding. Adding actual modeled harmonics here changes the signal representation, whereas increasing a real FFT's zero-padding length merely samples one windowed spectrum more densely.