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Allpass filter for lowpass to lowpass transformation

`[AllpassNum,AllpassDen] = allpasslp2lp(Wo,Wt)`

`[AllpassNum,AllpassDen] = allpasslp2lp(Wo,Wt)` returns
the numerator, `AllpassNum`, and the denominator, `AllpassDen`,
of the first-order allpass mapping filter for performing a real lowpass
to real lowpass frequency transformation. This transformation effectively
places one feature of an original filter, located originally at frequency `W`_{o},
at the required target frequency location, `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 lowpass to lowpass 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 and so on.

Lowpass to lowpass transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can change their position in a simple way by applying the lowpass to lowpass transformation.

Design the allpass filter changing the lowpass filter cutoff
frequency originally at `W`_{o}`=0.5` to `W`_{t}`=0.25`.
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:

Wo = 0.5; Wt = 0.25; [AllpassNum, AllpassDen] = allpasslp2lp(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');

As shown in the figure, `allpasslp2lp` generates
a mapping function that converts your prototype lowpass filter to
a target lowpass filter with different passband specifications.

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.

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., "Frequency transformations
for digital filters," *Electronics Letters*,
vol. 3, no. 11, pp. 487-489, November 1967.

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