## Documentation Center |

Allpass filter for lowpass to complex M-band transformation

`[AllpassNum,AllpassDen] = allpasslp2mbc(Wo,Wt)`

`[AllpassNum,AllpassDen] = allpasslp2mbc(Wo,Wt)` returns
the numerator, `AllpassNum`, and the denominator, `AllpassDen`,
of the `M`th-order allpass mapping filter for performing
a real lowpass to complex multipassband frequency transformation.
Parameter `M` is the number of times an original
feature is replicated in the target filter. 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}.

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, F_{1} and
F_{2}, with F_{1} preceding
F_{2}. Feature F_{1} will
still precede F_{2} after the transformation.
However, the distance between F_{1} and F_{2} will
not be the same before and after the transformation.

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; e.g., the stopband edge, the DC, the deep minimum in the stopband, or other ones.

This transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can be easily replicated at a number of required frequency locations without the need to design them again. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.

Design the allpass filter changing the real lowpass filter with
the cutoff frequency of `W`_{o}`=0.5` into
a complex multiband filter with band edges of `W`_{t}`=[-3+1:2:9]/10` precisely
defined:

Wo = 0.5; Wt = [-3+1:2:9]/10; [AllpassNum, AllpassDen] = allpasslp2mbc(Wo, Wt);

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

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

Plot the phase response normalized to π, which is in effect
the mapping function `W`_{o}`(W`_{t}`)`.
Please note that the transformation works in the same way for both
positive and negative frequencies:

plot(f/pi, angle(h)/pi, Wt, Wo, 'ro'); title('Mapping Function Wo(Wt)'); xlabel('New Frequency, Wt'); ylabel('Old Frequency, Wo');

In this example, the resulting mapping function converts real filters to multiband complex filters.

Variable | Description |
---|---|

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. |

AllpassNum | Numerator of the mapping filter |

AllpassDen | Denominator of the mapping filter |

Was this topic helpful?