[sdiy] Making a HPF using a LPF...??

[David G Dixon]

> You can make a high-pass filter by summing for a low-pass 
> filter, but... <snip>

Yes, but that's not now you would actually do the summing.  What you would
do for a cascaded two-stage LPF, for example, is to sum the second-stage
output with the input (or, more correctly, the input summed with the
feedback, which is the actual input to the first stage), and with twice the
inverted output of the first stage.  Hence, for a = 2 and b = 1,

2P LPF response:

HLP2(s) = 1/(s^2 + 2s + 1)

1P LPF response:

HLP1(s) = 1/(s + 1) = (s + 1)/(s + 1)^2 = (s + 1)/(s^2 + 2s + 1)

input (0P LPF response):

HLP0(s) = 1 = (s^2 + 2s + 1)/(s^2 + 2s + 1)

Therefore:

HLP2(s) - 2*HLP1(s) + HLP0(s) = (1 - 2s - 2 + s^2 + 2s + 1)/(s^2 + 2s + 1)

= s^2/(s^2 + 2s + 1) = HHP2(s) = 2P highpass response.

Again, this works very well, but only if the summing resistors are well
matched.