Chapter 8: Low-Pass

This demo shows the filtering of an input signal \(x[k]\) with a discrete low-pass filter of the form: $$h_\mathrm{LP}[k] = \begin{cases} \frac{\Omega_\mathrm g}{\pi}\cdot\mathrm{si}[(k-k_0)\Omega_\mathrm g] & 0\leq k\leq L-1\\ 0 & \mathrm{otherwise} \end{cases},$$ with the cut-off frequency \(\Omega_\mathrm g\in[0,\pi]\), length \(L\), and shift \(k_0=\left\lfloor\frac{L}{2}\right\rfloor\).

For smoothing the overshoot, a Hann-window (raised cosine) of length \(L\) can be used, which is defined as: $$\begin{align} w_\mathrm{Hann}[k] &= \frac{1}{2}\left[1-\cos\left(\frac{2\pi}{L-1}k\right)\right] \\ &= \sin^2\left(\frac{\pi}{L-1}k\right), \end{align}$$ for \(k=\{0,\dots,L-1\}\). The resulting low-pass signal is denoted as \(x_\mathrm{LP}[k]=x[k]\ast (h_\mathrm{LP}[k]\cdot w[k])\).

Input \(x[k]\) Fanfare Piano Cut-Off Frequency \(\Omega_\mathrm g/\pi\) : 0.2 Length \(L\)

16 32 64 128

Windowing (Hann-Window) Recompute

Original

Filtered