iirlp2bsc

Transform IIR lowpass filter to IIR complex bandstop filter

Syntax

[Num,Den,AllpassNum,AllpassDen] = iirlp2bsc(B,A,Wo,Wt)


Description

[Num,Den,AllpassNum,AllpassDen] = iirlp2bsc(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 first-order real lowpass to complex bandstop 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 the denominator specified by A.

This transformation effectively places one feature of an original filter, located at frequency -W_o, at the required target frequency location, W_t1, and the second feature, originally at +W_o, at the new location, W_t2. It is assumed that W_t2 is greater than W_t1. Additionally the transformation swaps passbands with stopbands in the target filter.

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 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.

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 at any place around the unit circle forming a pair of complex notches/resonators. This transformation can be used for designing bandstop filters for band attenuation or frequency equalizers, from the high-quality prototype lowpass filter.

Examples

Transform Lowpass Filter to Complex Bandstop Filter

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 complex bandstop filter by placing the cutoff frequencies of the prototype filter at –0.25π and 0.75π.

[num,den] = iirlp2bsc(b,a,0.5,[-0.25 0.75]);

Compare the magnitude responses of the filters using FVTool.

fvt = fvtool(b,a,num,den);
legend(fvt,'Prototype','Target')

Arguments

Variable	Description
B	Numerator of the prototype lowpass filter
A	Denominator of the prototype lowpass filter
Wo	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.
Wt	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.
Num	Numerator of the target filter
Den	Denominator of the target filter
AllpassNum	Numerator of the mapping filter
AllpassDen	Denominator of the mapping filter