Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

See Also

Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Analog Filters
       RLC Filter Analysis
          Impulse Response

Chapter Contents:

Search Introduction to Digital Filters

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Impulse Response

The impulse response is again the inverse Laplace transform of the transfer function. Expanding $ H(s)$ into a sum of complex one-pole sections,

$\displaystyle H(s) = 2\eta\cdot\frac{s}{s^2 + 2\eta s + \omega_0^2} = \frac{r_... ...r_2}{s-p_2} = \frac{(r_1+r_2)s - (r_1p_2 + r_2p_1)}{s^2-(p_1 + p_2) + p_1p_2}, $

where $ p_{1,2}\isdef -\eta \pm \sqrt{\eta^2 - \omega_0^2}$. Equating numerator coefficients gives

\begin{eqnarray*} r_1+r_2 &=& 2\eta \;\mathrel{=}\; \frac{1}{RC}\\ r_1p_2 + r_2p_1 &=& 0. \end{eqnarray*}

This pair of equations in two unknowns may be solved for $ r_1$ and $ r_2$. The impulse response is then

$\displaystyle h(t) = r_1 e^{p_1 t} u(t) + r_2 e^{p_2 t} u(t). $

Previous: Poles and Zeros
Next: Relating Pole Radius to Bandwidth

Order a Hardcopy of Introduction to Digital Filters

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )