help regarding impulse invariance method !

Started by April 13, 2009
Hi,
i hav just studied the impulse invariance method & bilinear transform.
In Impulse Invariance there is "many to one mapping" and hence aliasing
takes place.
while in Bilinear Transform there is "one to one mapping" and hence no
aliasing.
Actually what is the "one to one mapping" & "many to one mapping"?
how it is related to aliasing?

On Apr 13, 9:48&#2013266080;am, "nikkogta4" <klose0...@yahoo.co.in> wrote:
> > i hav just studied the impulse invariance method & bilinear transform. > In Impulse Invariance there is "many to one mapping" and hence aliasing > takes place. &#2013266080; > while in Bilinear Transform there is "one to one mapping" and hence no > aliasing. > Actually what is the "one to one mapping" & "many to one mapping"? > how it is related to aliasing?
i have not heard the term "many to one mapping" used regarding Impulse Invariant, but i know that it *does* have aliasing of the frequency response and why. the reason why is that the continuous-time analog impulse response is simply being sampled at the same sampling rate (1/ T) that the digital filter is supposed to be operating at (and scaled by T for dimensional/unit reasons) and that sampled impulse response becomes the discrete-time impulse response of the digital filter. whenever you sample in one domain (in this case, the time domain), you cause the spectrum of what you sampled to be repeatedly shifted (by multiples of 1/T) and overlapped and added in the other domain. that overlap and adding is what causes the aliased frequency response of an Impulse Invariant designed filter. the reason why Bilinear Transform (BLT) has "one to one mapping" is because, regarding the analog filter that is being transformed to a digital filter, for every frequency point in the frequency response of that analog filter, that frequency maps to exactly one frequency in the digital filter created using the BLT. that mapping is: f_d = 1/(pi*T) * arctan(pi*f_a*T) where f_a is the frequency in the analog frequency response and f_d is the frequency it maps to. what this means is every bump or dip or corner or *any* feature you see in the analog frequency response, you will see a corresponding feature in the digital filter frequency response, but at a (usually only slightly) different frequency. this is called the "frequency warping" property of the Bilinear Transform. r b-j