Transform IIR lowpass filter to IIR bandstop filter
[G,AllpassNum,AllpassDen] = iirlp2bs(Hd,Wo,Wt)
Hd is a
iirlp2bs(B,A,Wo,Wt) returns the numerator and denominator
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,
and the denominator,
AllpassDen, of the allpass
mapping filter. The prototype lowpass filter is given with a numerator
B and a denominator specified by
This transformation effectively places one feature of an original
filter, located at frequency -Wo, at the required
target frequency location, Wt1, and the second
feature, originally at
at the new location, Wt2. It is assumed that
Wt2 is greater than Wt1.
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 Wo and
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, F1 and F2, with F1 preceding F2. After the transformation feature F2 will precede F1 in the target filter. However, the distance between F1 and F2 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.
[G,AllpassNum,AllpassDen] = iirlp2bs(Hd,Wo,Wt) returns
a bandstop magnitude response. The coefficients
the allpass mapping filter for mapping the prototype filter frequency
the target frequencies vector
Wt. Note that in
Hd is a
with a lowpass magnitude response.
Design a prototype real IIR halfband filter using a standard elliptic approach:
[b, a] = ellip(3, 0.1, 30, 0.409);
Create the real bandstop filter by placing the cutoff frequencies
of the prototype filter at the band edge frequencies
[num, den] = iirlp2bs(b, a, 0.5, [0.25, 0.75]);
Verify the result by comparing the prototype filter with the target filter:
fvtool(b, a, num, den);
With both filters plotted in the figure, you see clearly the results of the transformation.
Numerator of the prototype lowpass filter
Denominator of the prototype lowpass filter
Frequency value to be transformed from the prototype filter
Desired frequency locations in the transformed target filter
Numerator of the target filter
Denominator of the target filter
Numerator of the mapping filter
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.