STFT Framing

Short-Time Fourier Transform Framing

The Short-Time Fourier Transform (STFT) analyzes how the frequency content of a signal changes over time. The key idea: divide the signal into short overlapping frames, then apply the DFT to each frame.

Framing

Given a signal $x$ of length $L$, extract frames of size $F$ with a hop size $H$:

$\text{frame}_i = x[i \cdot H : i \cdot H + F]$

Frames are extracted as long as the full frame fits within the signal. The number of frames is:

$n_{\text{frames}} = \lfloor (L - F) / H \rfloor + 1$

Overlap: When $H < F$, frames overlap. A 50% overlap means $H = F/2$.

STFT Frame Spectrum

For each frame, apply a window and compute the DFT magnitude:

$|X_i[k]| = |\text{DFT}(\text{frame}_i \cdot w)|$

The result is a 2D time-frequency representation called a spectrogram.

Example

Signal $[0,1,2,3,4,5,6,7]$, frame size $4$, hop $2$:

• Frame 0: $[0,1,2,3]$
• Frame 1: $[2,3,4,5]$
• Frame 2: $[4,5,6,7]$

Your Task

Implement:

• frame_signal(x, frame_size, hop) — returns list of frames (each a list of floats)
• stft_frame_spectrum(frame, window) — returns $|\text{DFT}(\text{frame} \cdot \text{window})|$

import math

def dft(x):
    N = len(x)
    X = []
    for k in range(N):
        val = 0 + 0j
        for n in range(N):
            val += x[n] * math.e ** (-2j * math.pi * k * n / N)
        X.append(val)
    return X

def frame_signal(x, frame_size, hop):
    frames = []
    i = 0
    while i + frame_size <= len(x):
        frames.append(x[i:i + frame_size])
        i += hop
    return frames

def stft_frame_spectrum(frame, window):
    windowed = [f * w for f, w in zip(frame, window)]
    return [abs(v) for v in dft(windowed)]