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Allpass filter for lowpass to bandstop transformation

`[AllpassNum,AllpassDen] = allpasslp2bs(Wo,Wt)`

`[AllpassNum,AllpassDen] = allpasslp2bs(Wo,Wt)` returns
the numerator, `AllpassNum`, and the denominator, `AllpassDen`,
of the second-order allpass mapping filter for performing a real lowpass
to real bandstop frequency transformation. 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}. This transformation
implements the "Nyquist Mobility," which means that the DC feature
stays at DC, but the Nyquist feature moves to a location dependent
on the selection of `W`_{o} and `W`_{t}.

Relative positions of other features of an original filter 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}.
After the transformation feature `F`_{2} will
precede `F`_{1} in the target
filter. 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 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.

Design the allpass filter changing the lowpass filter with cutoff
frequency at `W`_{o}`=0.5` to
the real bandstop filter with cutoff frequencies at `W`_{t1}`=0.25` and `W`_{t2}`=0.375`:

Wo = 0.5; Wt = [0.25, 0.375]; [AllpassNum, AllpassDen] = allpasslp2bs(Wo, Wt); [h, f] = freqz(AllpassNum, AllpassDen, 'whole'); plot(f/pi, abs(angle(h))/pi, Wt, Wo, 'ro'); title('Mapping Function Wo(Wt)'); xlabel('New Frequency, Wt'); ylabel('Old Frequency, Wo');

In the figure, you find the mapping filter function as determined by the example. Note the response is normalized to π:

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

Wo | Frequency value 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 0 and 1, with 1 corresponding to half the sample rate.

Constantinides, A.G., "Spectral transformations
for digital filters," *IEEE ^{®} Proceedings*,
vol. 117, no. 8, pp. 1585-1590, August 1970.

Nowrouzian, B. and A.G. Constantinides, "Prototype
reference transfer function parameters in the discrete-time frequency
transformations," *Proceedings 33rd Midwest Symposium
on Circuits and Systems*, Calgary, Canada, vol. 2, pp.
1078-1082, August 1990.

Nowrouzian, B. and L.T. Bruton, "Closed-form
solutions for discrete-time elliptic transfer functions," *Proceedings
of the 35th Midwest Symposium on Circuits and Systems*,
vol. 2, pp. 784-787, 1992.

Constantinides, A.G., "Design of bandpass
digital filters," *IEEE Proceedings*,
vol. 1, pp. 1129-1231, June 1969.

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