Filter Design Toolbox

allpassbpc2bpc

Return an allpass filter for complex bandpass transformation

Syntax

• [AllpassNum,AllpassDen] = allpassbpc2bpc(Wo,Wt)
  

Description

[AllpassNum,AllpassDen] = allpassbpc2bpc(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the first-order allpass mapping filter for performing a complex bandpass to complex bandpass frequency transformation. This transformation effectively places two features of an original filter, located at frequencies W_o1 and W_o2, at the required target frequency locations W_t1 and W_t2. It is assumed that W_t2 is greater than W_t1. In most of the cases the features selected for the transformation are the band edges of the filter passbands. 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.

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., complex notch filters or resonators can be repositioned at two distinct desired frequencies at any place around the unit circle. This is very attractive for adaptive systems.

Examples

Design the allpass filter changing the complex bandpass filter with the band edges originally at W_o1=0.2 and W_o2=0.4 to the new band edges of W_t1=0.3 and W_t2=0.6 precisely defined:

• Wo = [0.2, 0.4];
  Wt = [0.3, 0.6];
  [AllpassNum, AllpassDen] = allpassbpc2bpc(Wo, Wt);
  

Calculate the frequency response of the mapping filter in the full range:

• [ha, f] = freqz(AllpassNum, AllpassDen, 'whole');
  

Plot the phase response normalized to , which is in effect the mapping function W_o(W_t):

• plot(f/pi, angle(ha)/pi, Wt, Wo, 'ro');
  title('Mapping function Wo(Wt)');
  xlabel('New frequency, Wt');
  ylabel('Old frequency, Wo');
  

Arguments
Wo
   Frequency values to be transformed from the prototype filter

Wt
   Desired frequency locations in the transformed target filter

AllpassNum
   Numerator of the mapping filter

AllpassDen
   Denominator of the mapping filter

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

See Also
iirbpc2bpc, zpkbpc2bpc

adaptfilt.ufdaf

allpasslp2bp