Transform IIR lowpass filter to IIR complex M-band filter
[G,AllpassNum,AllpassDen] = iirlp2mbc(Hd,Wo,Wt)
Hd is a
iirlp2mbc(B,A,Wo,Wc) returns the numerator and denominator
of the target filter transformed from the real lowpass prototype by
Mth-order real lowpass to complex multibandpass
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
This transformation effectively places one feature of an original filter, located at frequency Wo, at the required target frequency locations, Wt1,...,WtM.
Relative positions of other features of an original filter do not change in the target filter. 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.
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.
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] = iirlp2mbc(Hd,Wo,Wt) returns
an IIR complex M-band filter frequency response. The coefficients
the allpass mapping filter for mapping the prototype filter frequency
the target frequencies vector
Wt. Note that in
Hd is a
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);
Now create a complex multiband filter with two passbands:
[num1, den1] = iirlp2mbc(b, a, 0.5, [2 4 6 8]/10);
Verify the result by comparing the prototype filter with the target filter:
fvtool(b, a, num1, den1);
You see in the figure that
the desired feature at 0.5 in the lowpass filter at four locations
in the multiband filter.
Numerator of the prototype lowpass filter.
Denominator of the prototype lowpass filter.
Frequency value to be transformed from the prototype filter. It 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.