zpklp2mbc (Filter Design Toolbox)

Zero-pole-gain lowpass to complex M-band frequency transformation

Syntax

[Z2,P2,K2,AllpassNum,AllpassDen] = zpklpmbc(Z,P,K,Wo,Wt)

Description

[Z2,P2,K2,AllpassNum,AllpassDen] = zpklpmbc(Z,P,K,Wo,Wt) returns zeros, Z₂, poles, P₂, and gain factor, K₂, of the target filter transformed from the real lowpass prototype by applying an Mth-order real lowpass to complex multibandpass frequency transformation.

It also returns the numerator, AllpassNum, and the denominator, AllpassDen, of the allpass mapping filter. The prototype lowpass filter is given with zeros, Z, poles, P, and gain factor, K.

This transformation effectively places one feature of an original filter, located at frequency W_o, at the required target frequency locations, W_t1,...,W_tM.

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, for example, the stopband edge, the DC, the deep minimum in the stopband, or other ones.

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.

This transformation can also be used for transforming other types of filters; e.g., to replicate notch filters and resonators at any required location.

Examples

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

[b, a] = ellip(3,0.1,30,0.409);
z = roots(b);
p = roots(a);
k = b(1);
[z1,p1,k1] = zpklp2mbc(z, p, k, 0.5, [2 4 6 8]/10, 'pass');
[z2,p2,k2] = zpklp2mbc(z, p, k, 0.5, [2 4 6 8]/10, 'stop');

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

fvtool(b, a, k1*poly(z1), poly(p1), k2*poly(z2), poly(p2));

Arguments
Z
Zeros of the prototype lowpass filter

P
Poles of the prototype lowpass filter

K
Gain factor 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.

Z2
Zeros of the target filter

P2
Poles of the target filter

K2
Gain factor of the target filter

AllpassNum
Numerator of the mapping filter

AllpassDen
Denominator of the mapping filter

See Also
zpkftransf, allpasslp2mbc, iirlp2mbc

zpklp2mb zpklp2xc