Spline Windows

A spline window of order $N$ is a repeated convolution of rectangular windows:

$$\begin{eqnarray*} w_{\text{Spline}(N)}(n) &=& (\underbrace{w_{R} \ast w_{R} \ast \dots \ast w_{R}}_{N+1})(n)\\ [10pt] &\leftrightarrow& W_{\text{Spline}(N)}(\omega) = \mathrm{asinc}^{N+1} \end{eqnarray*}$$

Special Cases:

• $N = 0\;\Rightarrow$ Rectangular (constant)
• $N = 1\;\Rightarrow$ Triangular (linear)
• $N = 2\;\Rightarrow$ Quadratic
• $N = 3\;\Rightarrow$ Cubic

Roll-Off Rate:

As $N$ increases, the window becomes smoother. $w_{\text{Spline}(N)}$ is $(N-1)$ -times continuously differentiable, and has roll-off rate $6(N+1)$ dB per octave.