Transform IIR lowpass filter to IIR bandpass filter


[Num,Den,AllpassNum,AllpassDen] = iirlp2bp(B,A,Wo,Wt)


[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.


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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);

Create a real bandpass filter by placing the cutoff frequencies of the prototype filter at 0.25π and 0.75π.

[num,den] = iirlp2bp(b,a,0.5,[0.25 0.75]);

Compare the magnitude responses of the filters using FVTool.

hvft = fvtool(b,a,num,den);



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.


[1] Constantinides, A.G., “Spectral transformations for digital filters,” IEEE® Proceedings, vol. 117, no. 8, pp. 1585-1590, August 1970.

[2] 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.

[3] Nowrouzian, B. and L.T. Bruton, “Closed-form solutions for discrete-time elliptic transfer functions,” Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, pp. 784-787, 1992.

[4] Constantinides, A.G., “Design of bandpass digital filters,' IEEE Proceedings, vol. 1, pp. 1129-1231, June 1969.

See Also


Introduced in R2011a