Transform IIR lowpass filter to IIR real N-point filter
iirlp2xn(B,A,Wo,Wt) returns the numerator and denominator
of the target filter transformed from the real lowpass prototype by
Nth-order real lowpass to real multipoint
frequency transformation, where
N is the number
of features being mapped. By default the DC feature is kept at its
iirlp2xn(B,A,Wo,Wt,Pass) allows you to specify an additional
Pass, which chooses between using the
"DC Mobility" and the "Nyquist Mobility." In the first case the Nyquist
feature stays at its original location and the DC feature is free
to move. In the second case the DC feature is kept at an original
frequency and the Nyquist feature is allowed to move.
It also returns the numerator,
and the denominator,
AllpassDen, of the allpass
mapping filter. The prototype lowpass filter is given with the numerator
B and the denominator specified by
N also specifies the number of
replicas of the prototype filter created around the unit circle after
the transformation. This transformation effectively places
of an original filter, located at frequencies Wo1,...,WoN,
at the required target frequency locations, Wt1,...,WtM.
Relative positions of other features of an original filter are the same in the target filter for the Nyquist mobility and are reversed for the DC mobility. For the Nyquist mobility this means that it is possible to select two features of an original filter, F1 and F2, with F1 preceding F2. Feature F1 will still precede F2 after the transformation. However, the distance between F1 and F2 will not be the same before and after the transformation. For DC mobility feature F2 will precede F1 after the transformation.
Choice of the feature subject to this transformation is not
restricted to the cutoff frequency of an original lowpass filter.
In general it is possible to select any feature; e.g., the stopband
edge, the DC, the deep minimum in the stopband, or other ones. The
only condition is that the features must be selected in such a way
that when creating
N bands around the unit circle,
there will be no band overlap.
This transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can be easily replicated at a number of required frequency locations. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.
Design a prototype real IIR lowpass elliptic filter with a gain of about –3 dB at 0.5π rad/sample.
[b,a] = ellip(3,0.1,30,0.409);
Transform the lowpass filter to an IIR real N-point filter. Compare the magnitude responses of the filters using FVTool.
[num,den] = iirlp2xn(b,a,[-0.5 0.5],[0.25 0.75]); hvft = fvtool(b,a,num,den); legend(hvft,'Prototype','Target')
Numerator of the prototype lowpass filter
Denominator of the prototype lowpass filter
Frequency values to be transformed from the prototype filter
Desired frequency locations in the transformed target filter
Numerator of the target filter
Denominator of the target filter
Numerator of the mapping filter
Denominator of the mapping filter
Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
 Cain, G.D., A. Krukowski and I. Kale, "High Order Transformations for Flexible IIR Filter Design," VII European Signal Processing Conference (EUSIPCO'94), vol. 3, pp. 1582-1585, Edinburgh, United Kingdom, September 1994.
 Krukowski, A., G.D. Cain and I. Kale, "Custom designed high-order frequency transformations for IIR filters," 38th Midwest Symposium on Circuits and Systems (MWSCAS'95), Rio de Janeiro, Brazil, August 1995.