[<<][math][>>][..]

Tue Mar 19 16:49:07 EDT 2013

## Partial fraction forms

http://www.wolframalpha.com/widgets/view.jsp?id=ec4a062bb304f88c2ba0b631d7acabbc
LP(s)
= pp' / (s - p) (s - p')
a b
= ------- + -------
s - p s - p'
=> as + bs = 0 => a = -b
- ap' - pb' = pp' => a (p - p') = pp'
or a = pp' / (p - p') = 2 |p|^2 / Im(p)
The wave form in the response is thus 90 degree shifted from a normal
exponential, of which the real part is a cosine and the imaginary part
a sine. I.e. the response is a sine wave, gradually rising to follow
the pulse. It doesn't jump immdiately, as would a high-pass filter.
HP(s)
= s^2 / (s - p) (s - p')
= 1 + p^2 / 2 Im(p) (s-p) + p'^2 / 2 Im(p) (s-p')
This has a direct path + a phase-shifted sinusoid.

[Reply][About]

[<<][math][>>][..]