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iirlp2bs

Transform IIR lowpass filter to IIR bandstop filter

Syntax

[Num,Den,AllpassNum,AllpassDen] = iirlp2bs(B,A,Wo,Wt)
[G,AllpassNum,AllpassDen] = iirlp2bs(Hd,Wo,Wt)

where Hd is a dfilt object

Description

[Num,Den,AllpassNum,AllpassDen] = iirlp2bs(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 bandstop 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 "Nyquist Mobility," which means that the DC feature stays at DC, but the Nyquist feature moves to a location dependent on the selection of Wo and Wts.

Relative positions of other features of an original filter 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. After the transformation feature F2 will precede F1 in the target filter. 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 bandstop transformation is not restricted only 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.

[G,AllpassNum,AllpassDen] = iirlp2bs(Hd,Wo,Wt) returns transformed dfilt object G with a bandstop magnitude 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.

Examples

Design a prototype real IIR halfband filter using a standard elliptic approach:

`[b, a] = ellip(3, 0.1, 30, 0.409);`

Create the real bandstop filter by placing the cutoff frequencies of the prototype filter at the band edge frequencies Wt1=0.25 and Wt2=0.75:

`[num, den] = iirlp2bs(b, a, 0.5, [0.25, 0.75]);`

Verify the result by comparing the prototype filter with the target filter:

`fvtool(b, a, num, den);`

With both filters plotted in the figure, you see clearly the results of the transformation.

Arguments

VariableDescription
B

Numerator of the prototype lowpass filter

A

Denominator of the prototype lowpass filter

Wo

Frequency value to be transformed from the prototype filter

Wt

Desired frequency locations in the transformed target filter

Num

Numerator of the target filter

Den

Denominator of the target filter

AllpassNum

Numerator of the mapping filter

AllpassDen

Denominator of the mapping filter

Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

References

Constantinides, A.G., "Spectral transformations for digital filters," IEEE® Proceedings, vol. 117, no. 8, pp. 1585-1590, August 1970.

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.

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.

Constantinides, A.G., "Design of bandpass digital filters," IEEE Proceedings, vol. 1, pp. 1129-1231, June 1969.