Transform IIR lowpass filter to IIR complex N-point filter
iirlp2xc(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 complex multipoint
It also returns the numerator,
and the denominator,
AllpassDen, of the allpass
mapping filter. The prototype lowpass filter is given with a numerator
B and a 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., a stopband
edge, 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
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 complex N-point filter. Compare the magnitude responses of the filters using FVTool.
[num,den] = iirlp2xc(b,a,[-0.5 0.5],[-0.25 0.25])
num = 1×4 complex 0.0643 - 0.0000i 0.0464 + 0.0000i 0.0464 + 0.0000i 0.0643 + 0.0000i
den = 1×4 complex 1.0000 + 0.0000i -1.6918 - 0.0000i 1.2340 + 0.0000i -0.3207 - 0.0000i
fvt = fvtool(b,a,num,den); legend(fvt,'Prototype','Target')
The target filter has complex coefficients and is indeed a bandpass filter.
Numerator of the prototype lowpass filter.
Denominator of the prototype lowpass filter.
Frequency values to be transformed from the prototype filter. They should be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
Desired frequency locations in the transformed target filter. They should be normalized to be between -1 and 1, with 1 corresponding to half the sample rate.
Numerator of the target filter.
Denominator of the target filter.
Numerator of the mapping filter.
Denominator of the mapping filter.