ELEC 310 formulas
complex functions
z = x + jy, f(z) = u(x,y) + jv(x,y) | (1) |
transfer function, ratio of polynomials with poles and zeros
(2) |
z-transform
(3) |
inverse z-transform,
c encloses all poles of
H(z) z^{k-1}
(4) |
geometric series
(5) |
(6) |
(7) |
basic inverse z-transform
use geometric series with w=pz^{-1} to obtain
transform pair
(8) |
Residue theorem
(9) |
first order pole
(10) |
(11) |
second order pole
(12) |
third order pole
(13) |
mth order pole
(14) |
frequency response (amplitude and phase), sampling rate f_{s}
(15) |
Discrete Fourier Transform (DFT) is frequency response sampled at N_{0} samples around the whole unit circle -f_{s}/2<f<f_{s}/2with frequency resolution (step size) f_{0}=f_{s}/N_{0}
H[r] | = | (16) | |
= | (17) | ||
= | (18) | ||
= | (19) | ||
= | (20) |
inverse DFT
(21) |