0af3a0ab-a3d2-4389-a4d8-b46e09a9cbba 16 4 5 decimal

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Analog/Digital Lowpass Butterworth Filter

T: sampling rate
................
................
fp: pasband edge frequency
fs: stopband edge frequency
G: minimum stopband attenuation T 0.001 : fmax 12 T * / : fp 90 : fs 100 : G 2 : 51 sys

Prewrap frequencies for the analog filter

ω.π 2 π * fp * :

ω.σ 2 π * fs * :

up 2 T / ω.π T 2 / * tan * :

us 2 T / ω.σ T 2 / * tan * :

up 581.0537

us 649.8394

x log x log10 :

Frequency normalization u us up / : 1.1184

Approximate order for Butterworth filter.
If the filter does not meet specs,
redefine 'N' for a higher order filter. N 1 G / (2^1 - log 2 u log * / Ceil Im :

N 15

'p' vector is sanity checked k 0 N 1 - (range p k 1 + el π 2 k * 1 + (* 2 N * / sin - i π 2 k * 1 + (* 2 N * / cos * + : for

optional => clear residual imaginary p 8 el 1 :

expansion is seemingly valid
XFR sanity checked MCD s H 1 s p ii el - / ii 1 N 0 - product :

i 1 - sqrt :

XFR x 0.5 1.5 0.55 range : j 1 x rows range xfr j el i x * (j el H abs eval : for x xfr augment 31 line :

region 1 0.2 1 0.707 u 0.707 u 0.2 1 0.2 52 mat :

sanity XFR 11 mat 0.5 1 0.55 1 0.6 1 0.65 1 0.7 1 0.75 0.9999 0.8 0.9994 0.85 0.9962 0.9 0.9795 0.95 0.9074 1 0.7071 1.05 0.4335 1.1 0.2328 1.15 0.122 1.2 0.0648 1.25 0.0352 1.3 0.0195 1.35 0.0111 1.4 0.0064 1.45 0.0038 1.5 0.0023 212 mat 11 mat

Normalized frequency reponse of the analog filter.

XFR region 2 1 sys

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Find the corresponding digital XFR using a bilinear transformation.

q 1 ω.π T 2 / * tan / :

z xfr q z 1 - z 1 + / * H :

data f 014 T * / 0.01 T / range : j 1 f rows range Y j el i 2 * π * f j el * T * exp xfr abs eval : for f Y augment 31 line :

plot digital data . 10 black plot : fp 0.707 o 8 red 15 mat fs 0.2 o 8 red 15 mat 31 sys :

Digital Butterworth filter

plot

There are a number of methods for converting functions from the continuous-time Laplace transform domain to the discrete-time z-transform domain. One of the most universal employs a bilinear transformation between the s-plane and z-plane.

s 2 T / 1 z 1 -^- 1 z 1 -^+ / (* ≡

... where T is the sampling rate.