How it works: The Discrete Fourier Transform (DFT) decomposes a window of N samples into N/2+1 frequency bins spaced Δf = fₛ/N Hz apart. X[k] = Σ_n x[n]·w[n]·e^−j2πkn/N where w[n] is the window function. A Rectangular window has sharp side-lobes; Hanning/Hamming reduce leakage; Blackman offers best leakage suppression at the cost of wider main lobe.
The Short-Time Fourier Transform (STFT) applies the DFT repeatedly on overlapping frames (hop size = N − overlap) and stacks the magnitude spectra into a coloured spectrogram. Time runs left-to-right, frequency bottom-to-top, and colour intensity encodes log magnitude.

Top-left: Time-domain waveform (2 cycles shown).   Bottom-left: DFT magnitude spectrum.   Right: STFT spectrogram (scrolling in real time). Yellow = loud, dark blue = silent.