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Zero-pole-gain lowpass to lowpass frequency transformation

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2lp(Z,P,K,Wo,Wt)`

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2lp(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 first-order real lowpass to real lowpass frequency mapping.
This transformation effectively places one feature of an original
filter, located at frequency W_{o}, at the required
target frequency location, W_{t}.

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`.

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, or other ones.

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 without designing them again.

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] = zpklp2lp(z, p, k, 0.5, 0.25);

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

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

Using `zpklp2lp` creates the desired half
band IIR filter with the transformed features that you specify in
the transformation function. This figure shows the results.

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," *IEE 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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