Transform IIR lowpass filter to IIR complex N-point filter
[G,AllpassNum,AllpassDen] = iirlp2xc(Hd,Wo,Wt)
where Hd is a dfilt object
[Num,Den,AllpassNum,AllpassDen] = iirlp2xc(B,A,Wo,Wt) returns the numerator and denominator vectors, Num and Den respectively, of the target filter transformed from the real lowpass prototype by applying an Nth-order real lowpass to complex multipoint frequency transformation.
It also returns the numerator, AllpassNum, and the denominator, AllpassDen, of the allpass mapping filter. The prototype lowpass filter is given with a numerator specified by B and a denominator specified by A.
Parameter N also specifies the number of replicas of the prototype filter created around the unit circle after the transformation. This transformation effectively places N features 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 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.
[G,AllpassNum,AllpassDen] = iirlp2xc(Hd,Wo,Wt) returns transformed dfilt object G with an IIR complex N-point filter frequency response. The coefficients AllpassNum and AllpassDen represent the allpass mapping filter for mapping the prototype filter frequency Wo and the target frequencies vector Wt. Note that in this syntax Hd is a dfilt object with a lowpass magnitude response.
Design a prototype real IIR halfband filter using a standard elliptic approach:
[b, a] = ellip(3, 0.1, 30, 0.409);
Create the complex bandpass filter from the real lowpass filter:
[num, den] = iirlp2xc(b, a, [-0.5 0.5], [-0.25 0.25]);
Verify the result by comparing the prototype filter with the target filter:
fvtool(b, a, num, den);
Reviewing the coefficients and the figure produced by the example shows that the target filter has complex coefficients and is indeed a bandpass filter as expected.
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.