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Allpass filter for lowpass to bandpass transformation

`[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt)`

`[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt)` returns
the numerator, `AllpassNum`, and the denominator, `AllpassDen`,
of the second-order allpass mapping filter for performing a real lowpass
to real bandpass 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 "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`_{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 bandpass 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.

Lowpass to bandpass transformation can also be used for transforming other types of filters; e.g., real notch filters or resonators can be doubled and repositioned at two distinct desired frequencies.

Design the allpass mapping filter changing the lowpass filter
with cutoff frequency at `W`_{o}`=0.5` to
the real–valued bandpass filter with cutoff frequencies at `W`_{t1}`0.25` and `W`_{t2}`=0.375`.

Compute the frequency response and 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, 0.375]; [AllpassNum, AllpassDen] = allpasslp2bp(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');

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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