Theory

The Hann window

ValkanPavlov Theory 18 April 2026

The Hann window is one of the most commonly used window functions in Digital Signal Processing, primarily because it introduces smooth transitions where the rectangular window produces abrupt discontinuities. Instead of weighting all samples equally, the Hann window applies a function that is maximal at the center of the interval and gradually tapers to zero at both ends. This makes it an excellent general-purpose window for applications where spectral leakage reduction is important but extreme suppression is not required.

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

The Rectangular Window

ValkanPavlov Theory 17 April 2026

The rectangular window is the most elementary case in digital signal processing (DSP). In fact, it can be thought of as "no window" – it simply selects a finite segment of the signal and leaves it unchanged within that interval, while setting all values outside it to zero. In other words, if you analyze a signal using the Discrete Fourier Transform (DFT), using a rectangular window means you are directly cutting the signal at the beginning and end of the segment under consideration.

\[ w(n) = \begin{cases} 1, & 0 \le n \le N-1 \\ 0, & \text{otherwise} \end{cases} \]

The problem comes from the fact that the DFT assumes this segment repeats periodically. If the beginning and end do not match smoothly, an artificial discontinuity appears between them. This discontinuity is not part of the real signal, but mathematically it is interpreted as the presence of additional frequencies. As a result, the energy that should be concentrated at a single frequency (e.g., in a sinusoid) gets spread over a wider range. This phenomenon is called spectral leakage and is the main drawback of the rectangular window.