Transform IIR lowpass filter to IIR bandpass filter
[G,AllpassNum,AllpassDen] = iirlp2bp(Hd,Wo,Wt)
where Hd is a dfilt object
[Num,Den,AllpassNum,AllpassDen] = iirlp2bp(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 a second-order real lowpass to real bandpass frequency mapping.
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.
This transformation effectively places one feature of an original filter, located at frequency -Wo, at the required target frequency location, Wt1, and the second feature, originally at +Wo, at the new location, Wt2. It is assumed that Wt2 is greater than Wt1. This transformation implements the "DC Mobility," meaning that the Nyquist feature stays at Nyquist, but the DC feature moves to a location dependent on the selection of Wts.
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 the lowpass to bandpass transformation is not restricted only to the cutoff frequency of an original lowpass filter. In general it is possible to select any feature: the stopband edge, the DC, the deep minimum in the stopband, or other ones.
Real lowpass to bandpass transformation can also be used for transforming other types of filters; e.g., real notch filters or resonators can be doubled and positioned at two distinct desired frequencies.
[G,AllpassNum,AllpassDen] = iirlp2bp(Hd,Wo,Wt) returns transformed dfilt object G with a real bandpass magnitude response. The coefficients AllpassNum and AllpassDen represent the allpass mapping filter for mapping the prototype filter frequency Wo and 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 real bandpass filter by placing the cutoff frequencies of the prototype filter at the band edge frequencies Wt1=0.25 and Wt2=0.75:
[num,den] = iirlp2bp(b,a,0.5,[0.25,0.75]);
Verify the result by comparing the prototype filter with the target filter:
You can compare the results in this figure to verify the transformation.
Numerator of the prototype lowpass filter
Denominator of the prototype lowpass filter
Frequency value 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.
Nowrouzian, B. and A.G. Constantinides, "Prototype reference transfer function parameters in the discrete-time frequency transformations," Proceedings 33rd Midwest Symposium on Circuits and Systems, Calgary, Canada, vol. 2, pp. 1078-1082, August 1990.
Constantinides, A.G., "Design of bandpass digital filters,' IEEE Proceedings, vol. 1, pp. 1129-1231, June 1969.