allpassshift
Allpass filter for real shift transformation
Syntax
[AllpassNum,AllpassDen] = allpassshift(Wo,Wt)
Description
[AllpassNum,AllpassDen] = allpassshift(Wo,Wt)
returns
the numerator, AllpassNum
, and the denominator, AllpassDen
,
of the second-order allpass mapping filter for performing a real frequency
shift transformation. This transformation places one selected feature
of an original filter, located at frequency W
o,
at the required target frequency location, W
t.
This transformation implements the “DC mobility,” which
means that the Nyquist feature stays at Nyquist, but the DC feature
moves to a location dependent on the selection of W
o and W
t.
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 the real shift 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 moved to a different frequency by applying a shift transformation. In such a way you can avoid designing the filter from the beginning.
Examples
Design the allpass filter precisely shifting one feature of
the lowpass filter originally at W
o=0.5
to
the new frequencies of W
t=0.25
:
Wo = 0.5; Wt = 0.25; [AllpassNum, AllpassDen] = allpassshift(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 Wo
(Wt)
.
Please note that the transformation works in the same way for both
positive and negative frequencies:
plot(f/pi, abs(angle(h))/pi, Wt, Wo, 'ro'); title('Mapping Function Wo(Wt)'); xlabel('New Frequency, Wt'); ylabel('Old Frequency, Wo');
Arguments
Variable | Description |
---|---|
Wo | Frequency value to be transformed from the prototype filter |
Wt | Desired frequency location in the transformed 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.
Version History
Introduced in R2011a