Filter Design Toolbox

iirlp2bs

Transform an IIR real lowpass filter to an IIR real bandstop filter frequency response

Syntax

• [Num,Den,AllpassNum,AllpassDen] = iirlp2bs(B,A,Wo,Wt)
  

Description

[Num,Den,AllpassNum,AllpassDen] = iirlp2bs(B,A,Wo,Wt) returns the numerator and denominator vectors, Num and Den respectively, 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 a numerator specified by B and a denominator specified by A.

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_ts.

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.

Examples

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 W_t1=0.25 and W_t2=0.75:

• [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);
  

Arguments
B
   Numerator of the prototype lowpass filter

A
   Denominator of the prototype lowpass filter

Wo
   Frequency value to be transformed from the prototype filter

Wt
   Desired frequency locations in the transformed target filter

Num
   Numerator of the target filter

Den
   Denominator 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.
  

See Also
iirftransf, allpasslp2bs, zpklp2bs

References

[1]  Constantinides, A.G., "Spectral transformations for digital filters," IEE Proceedings, vol. 117, no. 8, pp. 1585-1590, August 1970.

[2]  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.

[3]  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.

iirlp2bpc

iirlp2bsc