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Zero-pole-gain lowpass to bandstop frequency transformation

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2bs(Z,P,K,Wo,Wt)`

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2bs(Z,P,K,Wo,Wt)` returns zeros, `Z`_{2},
poles, `P`_{2}, and gain factor, `K`_{2},
of the target filter transformed from the real lowpass prototype by
applying a second-order real lowpass to real bandstop frequency mapping.

It also returns the numerator, `AllpassNum`,
and the denominator, `AllpassDen`, of the allpass
mapping filter. The prototype lowpass filter is given with zeros, `Z`,
poles, `P`, and gain factor, `K`.

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}s.

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 a prototype real IIR halfband filter using a standard elliptic approach:

[b, a] = ellip(3,0.1,30,0.409); z = roots(b); p = roots(a); k = b(1); [z2,p2,k2] = zpklp2bs(z, p, k, 0.5, [0.2 0.3]);

Verify the result by comparing the prototype filter with the target filter:

fvtool(b, a, k2*poly(z2), poly(p2));

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

Z | Zeros of the prototype lowpass filter |

P | Poles of the prototype lowpass filter |

K | Gain factor of the prototype lowpass filter |

Wo | Frequency value to be transformed from the prototype filter |

Wt | Desired frequency location in the transformed target filter |

Z2 | Zeros of the target filter |

P2 | Poles of the target filter |

K2 | Gain factor of the 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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