Windowing

Windowing

When computing the DFT of a finite signal, sharp edges at the start and end of the frame cause spectral leakage — energy from one frequency bin "leaks" into neighboring bins. A window function tapers the signal to zero at the edges, suppressing this effect.

The Hann Window

The Hann window (also called Hanning) is the most commonly used:

$w[n] = 0.5 \left(1 - \cos\!\left(\frac{2\pi n}{N-1}\right)\right), \quad n = 0, 1, \ldots, N-1$

Properties:

• $w[0] = w[N-1] = 0$ (tapers to zero at edges)
• $w[(N-1)/2] = 1.0$ (peak at center)
• Smooth roll-off reduces leakage dramatically

Example: 4-Point Hann Window

$w[0] = 0.5(1 - \cos(0)) = 0.0$ $w[1] = 0.5\left(1 - \cos\!\left(\frac{2\pi}{3}\right)\right) = 0.75$ $w[2] = 0.5\left(1 - \cos\!\left(\frac{4\pi}{3}\right)\right) = 0.75$ $w[3] = 0.5\left(1 - \cos(2\pi)\right) = 0.0$

Applying the Window

Windowing is an elementwise multiplication before the DFT:

$x_w[n] = x[n] \cdot w[n]$

Your Task

Implement:

• hann_window(N) — returns $[w[0], \ldots, w[N-1]]$
• apply_window(x, w) — returns elementwise product $[x[n] \cdot w[n]]$

import math

def hann_window(N):
    return [0.5 * (1 - math.cos(2 * math.pi * n / (N - 1))) for n in range(N)]

def apply_window(x, w):
    return [xi * wi for xi, wi in zip(x, w)]

print([round(v, 4) for v in hann_window(4)])  # [0.0, 0.75, 0.75, 0.0]