Low-pass filter
A low-pass filter passes what is below its corner frequency and cuts what is above. It smooths a demodulated signal, takes the carrier off a mixer's output, or limits the noise bandwidth.
Ports
One input, in, and one output, out.
The model
The filter is a Butterworth low-pass of order \(n\) (1 to 4), with corner \(f_c\). With \(s = i f/f_c\),
\[H(s) = \prod_{k=1}^{n} \frac{-p_k}{s - p_k}, \qquad p_k = \exp\!\left(i\pi\,\frac{2k + n - 1}{2n}\right),\]the poles evenly spaced on the left half of the unit circle. Its gain is maximally flat in the pass band:
\[\lvert H(f)\rvert = \frac{1}{\sqrt{1 + (f/f_c)^{2n}}},\]3 dB down at the corner, and falling at \(6n\) dB an octave (\(20n\) dB a decade) above it. At the corner each order turns the phase back by 45°, 90°, 135° or 180°. Tones are multiplied by \(H(f)\) and the noise density by \(\lvert H(f)\rvert^2\). It passes volts or amps alike.
Settings
| Field | Default | Meaning |
|---|---|---|
fc | 10000 | Corner (Hz) |
order | 1 | Order: 1 (1), 2 (2), 3 (3), 4 (4) |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Readouts
The panel gives its output in the steady state: the DC level, the five strongest tones (each one’s amplitude, rms and the phase of its cosine), and the noise, as a density at 1 kHz and in rms from 1 Hz to 1 GHz. Any notes, such as an input left unwired, are listed under it.
Assumptions
- An ideal Butterworth response: no component tolerances, no loss in the pass band and no noise of its own.