i hav just studied the impulse invariance method & bilinear transform.
In Impulse Invariance there is "many to one mapping" and hence aliasing
while in Bilinear Transform there is "one to one mapping" and hence no
Actually what is the "one to one mapping" & "many to one mapping"?
how it is related to aliasing?
Reply by robert bristow-johnson●April 13, 20092009-04-13
On Apr 13, 9:48�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. �
> while in Bilinear Transform there is "one to one mapping" and hence no
> 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.