STFT - Definition

STFT analyzes a signal 𝑥(𝑡) by multiplying it with a window function 𝑤(𝑡 − 𝜏) centered at time 𝜏, then taking the Fourier Transform:

Ordinary Fourier Transform analyzes a signal 𝑥(𝑡) with no window function:

Choosing Window Function

Different window shapes have different trade-offs in time vs frequency resolution:

Rectangular window	$w (t) = {10 ∣ t ∣ < T /2 otherwise$	simple, but causes spectral leakage (lots of side lobes in frequency)
Hamming window	$w (t) = 0.54 - 0.46 * cos (\frac{2 π t}{T - 1}), 0 \leq t \leq T - 1$	reduces leakage by tapering edges
Hann window	$w (t) = 0.5 * (1 - cos (\frac{2 π t}{T - 1} )), 0 \leq t \leq T - 1$	similar to Hamming
Gaussian window	$w (t) = e^{- π (t / σ)^{2}}$	smoothest, minimizes uncertainty principle when used the resulting STFT is called a Gabor Transform

Size (N)	Time span of frame	Frequency resolution Δf = Fs/N	Notes
256	~5.8 ms	~172 Hz/bin	Very fast, good for transients (drums, speech consonants), but poor pitch resolution
512	~11.6 ms	~86 Hz/bin	Balance for speech, can still track rapid events
1024	~23.2 ms	~43 Hz/bin	Common default, good mix for general audio/music
2048	~46.4 ms	~21 Hz/bin	Better frequency detail (notes, harmonics), worse timing
4096	~93 ms	~10.8 Hz/bin	High frequency precision, but smears fast events
8192	~186 ms	~5.4 Hz/bin	Super sharp frequency, very blurry in time — used in offline spectral analysis, not real-time