Impulse-Invariant Design 1.1 2005/02/13 20:12:13 US/Central 2005/04/24 16:10:48 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Jeffrey Silverman jsilv@rice.edu Kyle Clarkson kclarks@rice.edu alias pre-classical Pre-classical, adhoc-but-easy method of converting an analog prototype filter to a digital IIR filter. Does not preserve any optimality. Impulse invariance means that digital filter impulse response exactly equals samples of the analog prototype impulse response: n h n h a n T How is this done? The impulse response of a causal, stable analog filter is simply a sum of decaying exponentials: H a s b 0 b 1 s b 2 s 2 ... b p s p 1 a 1 s a 2 s 2 ... a p s p A 1 s s 1 A 2 s s 2 ... A p s s p which implies h a t A 1 s 1 t A 2 s 2 t ... A p s p t u t For impulse invariance, we desire h n h a n T A 1 s 1 n T A 2 s 2 n T ... A p s p n T u n Since A k s k T n u n A k z z s k T where z s k T , and H z k 1 p A k z z s k T where z k s k T . This technique is used occasionally in digital simulations of analog filters.What is the main problem/drawback with this design technique? Since it samples the non-bandlimited impulse response of the analog prototype filter, the frequency response aliases. This distorts the original analog frequency and destroys any optimal frequency properties in the resulting digital filter.