Prewarping

Type : explanation
References : Posted by robert bristow-johnson (better known as "rbj" )
Notes :
prewarping is simply recognizing the warping that the BLT introduces.
to determine frequency response, we evaluate the digital H(z) at
z=exp(j*w*T) and we evaluate the analog Ha(s) at s=j*W . the following
will confirm the jw to unit circle mapping and will show exactly what the
mapping is (this is the same stuff in the textbooks):

the BLT says: s = (2/T) * (z-1)/(z+1)

substituting: s = j*W = (2/T) * (exp(j*w*T) - 1) / (exp(j*w*T) + 1)

j*W = (2/T) * (exp(j*w*T/2) - exp(-j*w*T/2)) / (exp(j*w*T/2) + exp(-j*w*T/2))

= (2/T) * (j*2*sin(w*T/2)) / (2*cos(w*T/2))

= j * (2/T) * tan(w*T/2)

or

analog W = (2/T) * tan(w*T/2)

so when the real input frequency is w, the digital filter will behave with
the same amplitude gain and phase shift as the analog filter will have at a
hypothetical frequency of W. as w*T approaches pi (Nyquist) the digital
filter behaves as the analog filter does as W -> inf. for each degree of
freedom that you have in your design equations, you can adjust the analog
design frequency to be just right so that when the deterministic BLT
warping does its thing, the resultant warped frequency comes out just
right. for a simple LPF, you have only one degree of freedom, the cutoff
frequency. you can precompensate it so that the true cutoff comes out
right but that is it, above the cutoff, you will see that the LPF dives
down to -inf dB faster than an equivalent analog at the same frequencies.