9.4.3–9.4.4 Filters and windowed sinc

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §9.4.3–9.4.4. Rewritten in original wording. Continues 9.4b. Next: 9.4c labs.

Convolution = filtering

A linear filter maps input $f$ to output $g$ by convolution with an impulse response $h$. In frequency space that becomes a product. An ideal brick-wall lowpass is a rect in $\omega$ whose transform is a sinc. Pick $g=f*h$, the convolution theorem, and rect$\leftrightarrow$sinc among lookalikes.

$h(t)$ is the response to $\delta(t)$. Causality: $h(\tau)=0$ for effects before causes (no peeking into the future). Filter in frequency by multiplying spectra, then inverse-transform — often cheaper than a long time-domain convolution.

Analog RC sketches ($\tau=RC$) and the ideal rect$\leftrightarrow$sinc pair are part of the guided pick above: $|H_{\mathrm{lp}}|$ falls as $\omega$ grows; $|H_{\mathrm{hp}}|\to 1$ at large $\omega$ and vanishes at $\omega=0$. Phase comes from the imaginary pieces; magnitude carries the pass/stop story.

Digital taps

A delay line with coefficients $c_n$ implements

$$h(t)=\sum_{n=0}^{N} c_n\,\delta(t-n\tau) \quad\Rightarrow\quad H(\omega)=\sum_{n=0}^{N} c_n e^{-in\omega\tau},$$

and on a continuous (or sampled) input

$$g(t)=\sum_{n=0}^{N} c_n f(t-n\tau).$$

Knowing the $c_n$ is knowing the filter. A DFT itself is a bank of such frequency taps.

Windowed sinc (ideal lowpass, made practical)

Discrete sinc kernel (centered, then shifted to $t\ge 0$)

$$h[i]=\frac{\sin\bigl[2\pi\omega_c(i-M/2)\bigr]}{i-M/2},$$

truncated to $M+1$ taps ($M$ even). Truncation $\Rightarrow$ Gibbs ringing in $H(\omega)$. Soften the cut with a Hamming window

$$w[i]=0.54-0.46\cos\!\left(\frac{2\pi i}{M}\right),$$

and use $h[i]\,w[i]$. Choose $\omega_c$ as a fraction of the sampling rate; $M$ sets how sharp the transition band is. Same tool pretreats signals before a DFT to fight aliasing (§9.3).

Quiz

Filters and sinc

1 / 4

In the frequency domain, convolution f∗h becomes…

Pick an answer to see if you are right.

Next: 9.4c Filter & sinc labs.